Full-Information Item Factor Analysis (Multidimensional Item Response Theory)

mirt fits a maximum likelihood (or maximum a posteriori) factor analysis model to any mixture of dichotomous and polytomous data under the multidimensional item response theory paradigm using either Cai's (2010) Metropolis-Hastings Robbins-Monro (MHRM) algorithm, with an EM algorithm approach outlined by Bock and Aitkin (1981) using rectangular or quasi-Monte Carlo integration grids, or with the stochastic EM (i.e., the first two stages of the MH-RM algorithm). Unidimensional and multidimensional dominance/compensatory response models or unfolding/pairwise comparison models can be specified independently via the itemtype argument.

Usage

mirt(
  data,
  model = 1,
  itemtype = NULL,
  guess = 0,
  upper = 1,
  SE = FALSE,
  covdata = NULL,
  formula = NULL,
  itemdesign = NULL,
  item.formula = NULL,
  SE.type = "Oakes",
  method = "EM",
  optimizer = NULL,
  dentype = "Gaussian",
  pars = NULL,
  constrain = NULL,
  calcNull = FALSE,
  draws = 5000,
  survey.weights = NULL,
  quadpts = NULL,
  TOL = NULL,
  gpcm_mats = list(),
  grsm.block = NULL,
  rsm.block = NULL,
  monopoly.k = 1L,
  key = NULL,
  large = FALSE,
  GenRandomPars = FALSE,
  accelerate = "Ramsay",
  verbose = TRUE,
  solnp_args = list(),
  nloptr_args = list(),
  spline_args = list(),
  control = list(),
  technical = list(),
  ...
)

Arguments

data

a matrix or data.frame that consists of numerically ordered data, organized in the form of integers, with missing data coded as NA (to convert from an ordered factor data.frame see data.matrix)

model

a string to be passed (or an object returned from) mirt.model, declaring how the IRT model is to be estimated (loadings, constraints, priors, etc). For exploratory IRT models, a single numeric value indicating the number of factors to extract is also supported. Default is 1, indicating that a unidimensional model will be fit unless otherwise specified

itemtype

type of items to be modeled, declared as either a) a single value to be recycled for each item, b) a vector for each respective item, or c) if applicable, a matrix with columns equal to the number of items and rows equal to the number of latent classes. The NULL default assumes that the items follow a graded or 2PL structure, however they may be changed to the following:

'Rasch' - Rasch/partial credit model by constraining slopes to 1 and freely estimating the variance parameters (alternatively, can be specified by applying equality constraints to the slope parameters in 'gpcm' and '2PL'; Rasch, 1960)
'1PL', '2PL', '3PL', '3PLu', and '4PL' - 1-4 parameter logistic model, where 3PL estimates the lower asymptote only while 3PLu estimates the upper asymptote only (Lord and Novick, 1968; Lord, 1980). Note that specifying '1PL' will not automatically estimate the variance of the latent trait compared to the 'Rasch' type
'5PL' - 5 parameter logistic model to estimate asymmetric logistic response curves. Currently restricted to unidimensional models
'CLL' - complementary log-log link model. Currently restricted to unidimensional models
'ULL' - unipolar log-logistic model (Lucke, 2015). Note the use of this itemtype will automatically use a log-normal distribution for the latent traits
'graded' - graded response model (Samejima, 1969)
'grsm' - graded ratings scale model in the classical IRT parameterization (restricted to unidimensional models; Muraki, 1992)
'gpcm' and 'gpcmIRT' - generalized partial credit model in the slope-intercept and classical parameterization. 'gpcmIRT' is restricted to unidimensional models. Note that optional scoring matrices for 'gpcm' are available with the gpcm_mats input (Muraki, 1992)
'rsm' - Rasch rating scale model using the 'gpcmIRT' structure (unidimensional only; Andrich, 1978)
'nominal' - nominal response model (Bock, 1972)
'ideal' - dichotomous ideal point model (Maydeu-Olivares, 2006)
'ggum' - generalized graded unfolding model (Roberts, Donoghue, & Laughlin, 2000) and its multidimensional extension
'hcm' and 'ghcm' - (generalized) hyperbolic cosine model (Andrich and Luo, 1993; Andrich, 1996) for dichotomous or ordered polytomous items (see Luo, 2001)
'alm' and 'galm' - (generalized) absolute logistic model (Luo and Andrich, 2005) for dichotomous or ordered polytomous items
'sslm' and 'gsslm' - (generalized) simple squared logistic model (Andrich, 1988) for dichotomous or ordered polytomous items (see Luo, 2001)
'paralla' and 'gparalla' - (generalized) parallellogram analysis model (Hoijtink, 1990) for dichotomous or ordered polytomous items (see Luo, 2001)
'sequential' - multidimensional sequential response model (Tutz, 1990) in slope-intercept form
'Tutz' - same as the 'sequential' itemtype, except the slopes are fixed to 1 and the latent variance terms are freely estimated (similar to the 'Rasch' itemtype input)
'PC1PL', 'PC2PL', and 'PC3PL' - 1-3 parameter partially compensatory model. Note that constraining the slopes to be equal across items will also reduce the model to Embretson's (a.k.a. Whitely's) multicomponent model (1980), while for 'PC1PL' the slopes are fixed to 1 while the latent trait variance terms are estimated
'2PLNRM', '3PLNRM', '3PLuNRM', and '4PLNRM' - 2-4 parameter nested logistic model, where 3PLNRM estimates the lower asymptote only while 3PLuNRM estimates the upper asymptote only (Suh and Bolt, 2010)
'spline' - spline response model with the bs (default) or the ns function (Winsberg, Thissen, and Wainer, 1984)
'monopoly' - monotonic polynomial model for unidimensional tests for dichotomous and polytomous response data (Falk and Cai, 2016)

Additionally, user defined item classes can also be defined using the createItem function

guess

fixed pseudo-guessing parameters. Can be entered as a single value to assign a global guessing parameter or may be entered as a numeric vector corresponding to each item

upper

fixed upper bound parameters for 4-PL model. Can be entered as a single value to assign a global guessing parameter or may be entered as a numeric vector corresponding to each item

SE

logical; estimate the standard errors by computing the parameter information matrix? See SE.type for the type of estimates available

covdata

a data.frame of data used for latent regression models

formula

an R formula (or list of formulas) indicating how the latent traits can be regressed using external covariates in covdata. If a named list of formulas is supplied (where the names correspond to the latent trait names in model) then specific regression effects can be estimated for each factor. Supplying a single formula will estimate the regression parameters for all latent traits by default

itemdesign

a data.frame with rows equal to the number of items and columns containing any item-design effects. If items should be included in the design structure (i.e., should be left in their canonical structure) then fewer rows can be used, however the rownames must be defined and matched with colnames in the data input. The item design matrix is constructed with the use of item.formula. Providing this input will fix the associated 'd' intercepts to 0, where applicable

item.formula

an R formula used to specify any intercept decomposition (e.g., the LLTM; Fischer, 1983). Note that only the right-hand side of the formula is required for compensatory models.

For non-compensatory itemtypes (e.g., 'PC1PL') the formula must include the name of the latent trait in the left hand side of the expression to indicate which of the trait specification should have their intercepts decomposed (see MLTM; Embretson, 1984)

SE.type

type of estimation method to use for calculating the parameter information matrix for computing standard errors and wald tests. Can be:

'Richardson', 'forward', or 'central' for the numerical Richardson, forward difference, and central difference evaluation of observed Hessian matrix
'crossprod' and 'Louis' for standard error computations based on the variance of the Fisher scores as well as Louis' (1982) exact computation of the observed information matrix. Note that Louis' estimates can take a long time to obtain for large sample sizes and long tests
'sandwich' for the sandwich covariance estimate based on the 'crossprod' and 'Oakes' estimates (see Chalmers, 2018, for details)
'sandwich.Louis' for the sandwich covariance estimate based on the 'crossprod' and 'Louis' estimates
'Oakes' for Oakes' (1999) method using a central difference approximation (see Chalmers, 2018, for details)
'SEM' for the supplemented EM (disables the accelerate option automatically; EM only)
'Fisher' for the expected information, 'complete' for information based on the complete-data Hessian used in EM algorithm
'MHRM' and 'FMHRM' for stochastic approximations of observed information matrix based on the Robbins-Monro filter or a fixed number of MHRM draws without the RM filter. These are the only options supported when method = 'MHRM'
'numerical' to obtain the numerical estimate from a call to optim when method = 'BL'

Note that both the 'SEM' method becomes very sensitive if the ML solution has has not been reached with sufficient precision, and may be further sensitive if the history of the EM cycles is not stable/sufficient for convergence of the respective estimates. Increasing the number of iterations (increasing NCYCLES and decreasing TOL, see below) will help to improve the accuracy, and can be run in parallel if a mirtCluster object has been defined (this will be used for Oakes' method as well). Additionally, inspecting the symmetry of the ACOV matrix for convergence issues by passing technical = list(symmetric = FALSE) can be helpful to determine if a sufficient solution has been reached

method

a character object specifying the estimation algorithm to be used. The default is 'EM', for the standard EM algorithm with fixed quadrature, 'QMCEM' for quasi-Monte Carlo EM estimation, or 'MCEM' for Monte Carlo EM estimation. The option 'MHRM' may also be passed to use the MH-RM algorithm, 'SEM' for the Stochastic EM algorithm (first two stages of the MH-RM stage using an optimizer other than a single Newton-Raphson iteration), and 'BL' for the Bock and Lieberman approach (generally not recommended for longer tests).

The 'EM' is generally effective with 1-3 factors, but methods such as the 'QMCEM', 'MCEM', 'SEM', or 'MHRM' should be used when the dimensions are 3 or more. Note that when the optimizer is stochastic the associated SE.type is automatically changed to SE.type = 'MHRM' by default to avoid the use of quadrature

optimizer

a character indicating which numerical optimizer to use. By default, the EM algorithm will use the 'BFGS' when there are no upper and lower bounds box-constraints and 'nlminb' when there are.

Other options include the Newton-Raphson ('NR'), which can be more efficient than the 'BFGS' but not as stable for more complex IRT models (such as the nominal or nested logit models) and the related 'NR1' which is also the Newton-Raphson but consists of only 1 update that has been coupled with RM Hessian (only applicable when the MH-RM algorithm is used). The MH-RM algorithm uses the 'NR1' by default, though currently the 'BFGS', 'L-BFGS-B', and 'NR' are also supported with this method (with fewer iterations by default) to emulate stochastic EM updates. As well, the 'Nelder-Mead' and 'SANN' estimators are available, but their routine use generally is not required or recommended.

Additionally, estimation subroutines from the Rsolnp and nloptr packages are available by passing the arguments 'solnp' and 'nloptr', respectively. This should be used in conjunction with the solnp_args and nloptr_args specified below. If equality constraints were specified in the model definition only the parameter with the lowest parnum in the pars = 'values' data.frame is used in the estimation vector passed to the objective function, and group hyper-parameters are omitted. Equality an inequality functions should be of the form function(p, optim_args), where optim_args is a list of internally parameters that largely can be ignored when defining constraints (though use of browser() here may be helpful)

dentype

type of density form to use for the latent trait parameters. Current options include

'Gaussian' (default) assumes a multivariate Gaussian distribution with an associated mean vector and variance-covariance matrix
'empiricalhist' or 'EH' estimates latent distribution using an empirical histogram described by Bock and Aitkin (1981). Only applicable for unidimensional models estimated with the EM algorithm. For this option, the number of cycles, TOL, and quadpts are adjusted accommodate for less precision during estimation (namely: TOL = 3e-5, NCYCLES = 2000, quadpts = 121)
'empiricalhist_Woods' or 'EHW' estimates latent distribution using an empirical histogram described by Bock and Aitkin (1981), with the same specifications as in dentype = 'empiricalhist', but with the extrapolation-interpolation method described by Woods (2007). NOTE: to improve stability in the presence of extreme response styles (i.e., all highest or lowest in each item) the technical option zeroExtreme = TRUE may be required to down-weight the contribution of these problematic patterns
'Davidian-#' estimates semi-parametric Davidian curves described by Woods and Lin (2009), where the # placeholder represents the number of Davidian parameters to estimate (e.g., 'Davidian-6' will estimate 6 smoothing parameters). By default, the number of quadpts is increased to 121, and this method is only applicable for unidimensional models estimated with the EM algorithm

Note that when itemtype = 'ULL' then a log-normal(0,1) density is used to support the unipolar scaling

pars

a data.frame with the structure of how the starting values, parameter numbers, estimation logical values, etc, are defined. The user may observe how the model defines the values by using pars = 'values', and this object can in turn be modified and input back into the estimation with pars = mymodifiedpars

constrain

a list of user declared equality constraints. To see how to define the parameters correctly use pars = 'values' initially to see how the parameters are labeled. To constrain parameters to be equal create a list with separate concatenated vectors signifying which parameters to constrain. For example, to set parameters 1 and 5 equal, and also set parameters 2, 6, and 10 equal use constrain = list(c(1,5), c(2,6,10)). Constraints can also be specified using the mirt.model syntax (recommended)

calcNull

logical; calculate the Null model for additional fit statistics (e.g., TLI)? Only applicable if the data contains no NA's and the data is not overly sparse

draws

the number of Monte Carlo draws to estimate the log-likelihood for the MH-RM algorithm. Default is 5000

survey.weights

a optional numeric vector of survey weights to apply for each case in the data (EM estimation only). If not specified, all cases are weighted equally (the standard IRT approach). The sum of the survey.weights must equal the total sample size for proper weighting to be applied

quadpts

number of quadrature points per dimension (must be larger than 2). By default the number of quadrature uses the following scheme: switch(as.character(nfact), '1'=61, '2'=31, '3'=15, '4'=9, '5'=7, 3). However, if the method input is set to 'QMCEM' and this argument is left blank then the default number of quasi-Monte Carlo integration nodes will be set to 5000 in total

TOL

convergence threshold for EM or MH-RM; defaults are .0001 and .001. If SE.type = 'SEM' and this value is not specified, the default is set to 1e-5. To evaluate the model using only the starting values pass TOL = NaN, and to evaluate the starting values without the log-likelihood pass TOL = NA

gpcm_mats

a list of matrices specifying how the scoring coefficients in the (generalized) partial credit model should be constructed. If omitted, the standard gpcm format will be used (i.e., seq(0, k, by = 1) for each trait). This input should be used if traits should be scored different for each category (e.g., matrix(c(0:3, 1,0,0,0), 4, 2) for a two-dimensional model where the first trait is scored like a gpcm, but the second trait is only positively indicated when the first category is selected). Can be used when itemtypes are 'gpcm' or 'Rasch', but only when the respective element in gpcm_mats is not NULL

grsm.block

an optional numeric vector indicating where the blocking should occur when using the grsm, NA represents items that do not belong to the grsm block (other items that may be estimated in the test data). For example, to specify two blocks of 3 with a 2PL item for the last item: grsm.block = c(rep(1,3), rep(2,3), NA). If NULL the all items are assumed to be within the same group and therefore have the same number of item categories

rsm.block

same as grsm.block, but for 'rsm' blocks

monopoly.k

a vector of values (or a single value to repeated for each item) which indicate the degree of the monotone polynomial fitted, where the monotone polynomial corresponds to monopoly.k * 2 + 1 (e.g., monopoly.k = 2 fits a 5th degree polynomial). Default is monopoly.k = 1, which fits a 3rd degree polynomial

key

a numeric vector of the response scoring key. Required when using nested logit item types, and must be the same length as the number of items used. Items that are not nested logit will ignore this vector, so use NA in item locations that are not applicable

large

a logical indicating whether unique response patterns should be obtained prior to performing the estimation so as to avoid repeating computations on identical patterns. The default TRUE provides the correct degrees of freedom for the model since all unique patterns are tallied (typically only affects goodness of fit statistics such as G2, but also will influence nested model comparison methods such as anova(mod1, mod2)), while FALSE will use the number of rows in data as a placeholder for the total degrees of freedom. As such, model objects should only be compared if all flags were set to TRUE or all were set to FALSE

Alternatively, if the collapse table of frequencies is desired for the purpose of saving computations (i.e., only computing the collapsed frequencies for the data onte-time) then a character vector can be passed with the arguement large = 'return' to return a list of all the desired table information used by mirt. This list object can then be reused by passing it back into the large argument to avoid re-tallying the data again (again, useful when the dataset are very large and computing the tabulated data is computationally burdensome). This strategy is shown below:

Compute organized data: e.g., internaldat <- mirt(Science, 1, large = 'return')
Pass the organized data to all estimation functions: e.g., mod <- mirt(Science, 1, large = internaldat)

GenRandomPars

logical; generate random starting values prior to optimization instead of using the fixed internal starting values?

accelerate

a character vector indicating the type of acceleration to use. Default is 'Ramsay', but may also be 'squarem' for the SQUAREM procedure (specifically, the gSqS3 approach) described in Varadhan and Roldand (2008). To disable the acceleration, pass 'none'

verbose

logical; print observed- (EM) or complete-data (MHRM) log-likelihood after each iteration cycle? Default is TRUE

solnp_args

a list of arguments to be passed to the solnp::solnp() function for equality constraints, inequality constraints, etc

nloptr_args

a list of arguments to be passed to the nloptr::nloptr() function for equality constraints, inequality constraints, etc

spline_args

a named list of lists containing information to be passed to the bs (default) and ns for each spline itemtype. Each element must refer to the name of the itemtype with the spline, while the internal list names refer to the arguments which are passed. For example, if item 2 were called 'read2', and item 5 were called 'read5', both of which were of itemtype 'spline' but item 5 should use the ns form, then a modified list for each input might be of the form:

spline_args = list(read2 = list(degree = 4), read5 = list(fun = 'ns', knots = c(-2, 2)))

This code input changes the bs() splines function to have a degree = 4 input, while the second element changes to the ns() function with knots set a c(-2, 2)

control

a list passed to the respective optimizers (i.e., optim(), nlminb(), etc). Additional arguments have been included for the 'NR' optimizer: 'tol' for the convergence tolerance in the M-step (default is TOL/1000), while the default number of iterations for the Newton-Raphson optimizer is 50 (modified with the 'maxit' control input)

technical

a list containing lower level technical parameters for estimation. May be:

NCYCLES: maximum number of EM or MH-RM cycles; defaults are 500 and 2000
MAXQUAD: maximum number of quadratures, which you can increase if you have more than 4GB or RAM on your PC; default 20000
theta_lim: range of integration grid for each dimension; default is c(-6, 6). Note that when itemtype = 'ULL' a log-normal distribution is used and the range is change to c(.01, and 6^2), where the second term is the square of the theta_lim input instead
set.seed: seed number used during estimation. Default is 12345
SEtol: standard error tolerance criteria for the S-EM and MHRM computation of the information matrix. Default is 1e-3
symmetric: logical; force S-EM/Oakes information matrix estimates to be symmetric? Default is TRUE so that computation of standard errors are more stable. Setting this to FALSE can help to detect solutions that have not reached the ML estimate
SEM_window: ratio of values used to define the S-EM window based on the observed likelihood differences across EM iterations. The default is c(0, 1 - SEtol), which provides nearly the very full S-EM window (i.e., nearly all EM cycles used). To use the a smaller SEM window change the window to to something like c(.9, .999) to start at a point farther into the EM history
warn: logical; include warning messages during estimation? Default is TRUE
message: logical; include general messages during estimation? Default is TRUE
customK: a numeric vector used to explicitly declare the number of response categories for each item. This should only be used when constructing mirt model for reasons other than parameter estimation (such as to obtain factor scores), and requires that the input data all have 0 as the lowest category. The format is the same as the extract.mirt(mod, 'K') slot in all converged models
customPriorFun: a custom function used to determine the normalized density for integration in the EM algorithm. Must be of the form function(Theta, Etable){...}, and return a numeric vector with the same length as number of rows in Theta. The Etable input contains the aggregated table generated from the current E-step computations. For proper integration, the returned vector should sum to 1 (i.e., normalized). Note that if using the Etable it will be NULL on the first call, therefore the prior will have to deal with this issue accordingly
zeroExtreme: logical; assign extreme response patterns a survey.weight of 0 (formally equivalent to removing these data vectors during estimation)? When dentype = 'EHW', where Woods' extrapolation is utilized, this option may be required if the extrapolation causes expected densities to tend towards positive or negative infinity. The default is FALSE
customTheta: a custom Theta grid, in matrix form, used for integration. If not defined, the grid is determined internally based on the number of quadpts
nconstrain: same specification as the constrain list argument, however imposes a negative equality constraint instead (e.g., $a12 = -a21$, which is specified as nconstrain = list(c(12, 21))). Note that each specification in the list must be of length 2, where the second element is taken to be -1 times the first element
delta: the deviation term used in numerical estimates when computing the ACOV matrix with the 'forward' or 'central' numerical approaches, as well as Oakes' method with the Richardson extrapolation. Default is 1e-5
parallel: logical; use the parallel cluster defined by mirtCluster? Default is TRUE
storeEMhistory: logical; store the iteration history when using the EM algorithm? Default is FALSE. When TRUE, use extract.mirt to extract
internal_constraints: logical; include the internal constraints when using certain IRT models (e.g., 'grsm' itemtype). Disable this if you want to use special optimizers such as the solnp. Default is TRUE
gain: a vector of two values specifying the numerator and exponent values for the RM gain function $(val1 / cycle)^val2$. Default is c(0.10, 0.75)
BURNIN: number of burn in cycles (stage 1) in MH-RM; default is 150
SEMCYCLES: number of SEM cycles (stage 2) in MH-RM; default is 100
MHDRAWS: number of Metropolis-Hasting draws to use in the MH-RM at each iteration; default is 5
MHcand: a vector of values used to tune the MH sampler. Larger values will cause the acceptance ratio to decrease. One value is required for each group in unconditional item factor analysis (mixedmirt() requires additional values for random effect). If null, these values are determined internally, attempting to tune the acceptance of the draws to be between .1 and .4
MHRM_SE_draws: number of fixed draws to use when SE=TRUE and SE.type = 'FMHRM' and the maximum number of draws when SE.type = 'MHRM'. Default is 2000
MCEM_draws: a function used to determine the number of quadrature points to draw for the 'MCEM' method. Must include one argument which indicates the iteration number of the EM cycle. Default is function(cycles) 500 + (cycles - 1)*2, which starts the number of draws at 500 and increases by 2 after each full EM iteration
info_if_converged: logical; compute the information matrix when using the MH-RM algorithm only if the model converged within a suitable number of iterations? Default is TRUE
logLik_if_converged: logical; compute the observed log-likelihood when using the MH-RM algorithm only if the model converged within a suitable number of iterations? Default is TRUE
keep_vcov_PD: logical; attempt to keep the variance-covariance matrix of the latent traits positive definite during estimation in the EM algorithm? This generally improves the convergence properties when the traits are highly correlated. Default is TRUE

...

additional arguments to be passed

Value

function returns an object of class SingleGroupClass (SingleGroupClass-class)

Details

Models containing 'explanatory' person or item level predictors can only be included by using the mixedmirt function, though latent regression models can be fit using the formula input in this function. Tests that form a two-tier or bi-factor structure should be estimated with the bfactor function, which uses a dimension reduction EM algorithm for modeling item parcels. Multiple group analyses (useful for DIF and DTF testing) are also available using the multipleGroup function.

Confirmatory and Exploratory IRT

Specification of the confirmatory item factor analysis model follows many of the rules in the structural equation modeling framework for confirmatory factor analysis. The variances of the latent factors are automatically fixed to 1 to help facilitate model identification. All parameters may be fixed to constant values or set equal to other parameters using the appropriate declarations. Confirmatory models may also contain 'explanatory' person or item level predictors, though including predictors is currently limited to the mixedmirt function.

When specifying a single number greater than 1 as the model input to mirt an exploratory IRT model will be estimated. Rotation and target matrix options are available if they are passed to generic functions such as summary-method and fscores. Factor means and variances are fixed to ensure proper identification.

If the model is an exploratory item factor analysis estimation will begin by computing a matrix of quasi-polychoric correlations. A factor analysis with nfact is then extracted and item parameters are estimated by $a_{ij} = f_{ij}/u_j$, where $f_{ij}$ is the factor loading for the jth item on the ith factor, and $u_j$ is the square root of the factor uniqueness, $\sqrt{1 - h_j^2}$. The initial intercept parameters are determined by calculating the inverse normal of the item facility (i.e., item easiness), $q_j$, to obtain $d_j = q_j / u_j$. A similar implementation is also used for obtaining initial values for polytomous items.

A note on upper and lower bound parameters

Internally the $g$ and $u$ parameters are transformed using a logit transformation ($log(x/(1-x))$), and can be reversed by using $1 / (1 + exp(-x))$ following convergence. This also applies when computing confidence intervals for these parameters, and is done so automatically if coef(mod, rawug = FALSE).

As such, when applying prior distributions to these parameters it is recommended to use a prior that ranges from negative infinity to positive infinity, such as the normally distributed prior via the 'norm' input (see mirt.model).

Convergence for quadrature methods

Unrestricted full-information factor analysis is known to have problems with convergence, and some items may need to be constrained or removed entirely to allow for an acceptable solution. As a general rule dichotomous items with means greater than .95, or items that are only .05 greater than the guessing parameter, should be considered for removal from the analysis or treated with prior parameter distributions. The same type of reasoning is applicable when including upper bound parameters as well. For polytomous items, if categories are rarely endorsed then this will cause similar issues. Also, increasing the number of quadrature points per dimension, or using the quasi-Monte Carlo integration method, may help to stabilize the estimation process in higher dimensions. Finally, solutions that are not well defined also will have difficulty converging, and can indicate that the model has been misspecified (e.g., extracting too many dimensions).

Convergence for MH-RM method

For the MH-RM algorithm, when the number of iterations grows very high (e.g., greater than 1500) or when Max Change = .2500 values are repeatedly printed to the console too often (indicating that the parameters were being constrained since they are naturally moving in steps greater than 0.25) then the model may either be ill defined or have a very flat likelihood surface, and genuine maximum-likelihood parameter estimates may be difficult to find. Standard errors are computed following the model convergence by passing SE = TRUE, to perform an addition MH-RM stage but treating the maximum-likelihood estimates as fixed points.

Additional helper functions

Additional functions are available in the package which can be useful pre- and post-estimation. These are:

mirt.model: Define the IRT model specification use special syntax. Useful for defining between and within group parameter constraints, prior parameter distributions, and specifying the slope coefficients for each factor
coef-method: Extract raw coefficients from the model, along with their standard errors and confidence intervals
summary-method: Extract standardized loadings from model. Accepts a rotate argument for exploratory item response model
anova-method: Compare nested models using likelihood ratio statistics as well as information criteria such as the AIC and BIC
residuals-method: Compute pairwise residuals between each item using methods such as the LD statistic (Chen & Thissen, 1997), as well as response pattern residuals
plot-method: Plot various types of test level plots including the test score and information functions and more
itemplot: Plot various types of item level plots, including the score, standard error, and information functions, and more
createItem: Create a customized itemtype that does not currently exist in the package
imputeMissing: Impute missing data given some computed Theta matrix
fscores: Find predicted scores for the latent traits using estimation methods such as EAP, MAP, ML, WLE, and EAPsum
wald: Compute Wald statistics follow the convergence of a model with a suitable information matrix
M2: Limited information goodness of fit test statistic based to determine how well the model fits the data
itemfit and personfit: Goodness of fit statistics at the item and person levels, such as the S-X2, infit, outfit, and more
boot.mirt: Compute estimated parameter confidence intervals via the bootstrap methods
mirtCluster: Define a cluster for the package functions to use for capitalizing on multi-core architecture to utilize available CPUs when possible. Will help to decrease estimation times for tasks that can be run in parallel

IRT Models

The parameter labels use the follow convention, here using two factors and $K$ as the total number of categories (using $k$ for specific category instances).

Rasch

Only one intercept estimated, and the latent variance of $\theta$ is freely estimated. If the data have more than two categories then a partial credit model is used instead (see 'gpcm' below). $$P(x = 1|\theta, d) = \frac{1}{1 + exp(-(\theta + d))}$$

1-4PL

Depending on the model $u$ may be equal to 1 (e.g., 3PL), $g$ may be equal to 0 (e.g., 2PL), or the as may be fixed to 1 (e.g., 1PL). $$P(x = 1|\theta, \psi) = g + \frac{(u - g)}{ 1 + exp(-(a_1 * \theta_1 + a_2 * \theta_2 + d))}$$

5PL

Currently restricted to unidimensional models $$P(x = 1|\theta, \psi) = g + \frac{(u - g)}{ 1 + exp(-(a_1 * \theta_1 + d))^S}$$ where $S$ allows for asymmetry in the response function and is transformation constrained to be greater than 0 (i.e., log(S) is estimated rather than S)

CLL

Complementary log-log model (see Shim, Bonifay, and Wiedermann, 2022) $$P(x = 1|\theta, b) = 1 - exp(-exp(\theta - b))$$ Currently restricted to unidimensional dichotomous data.

graded

ULL

The unipolar log-logistic model (ULL; Lucke, 2015) is defined the same as the graded response model, however $$P(x \le k | \theta, \psi) = \frac{\lambda_k\theta^\eta}{1 + \lambda_k\theta^\eta}$$. Internally the $\lambda$ parameters are exponentiated to keep them positive, and should therefore the reported estimates should be interpreted in log units

grsm

A more constrained version of the graded model where graded spacing is equal across item blocks and only adjusted by a single 'difficulty' parameter (c) while the latent variance of $\theta$ is freely estimated (see Muraki, 1990 for this exact form). This is restricted to unidimensional models only.

gpcm/nominal

For the gpcm the $d$ values are treated as fixed and ordered values from $0:(K-1)$ (in the nominal model $d_0$ is also set to 0). Additionally, for identification in the nominal model $ak_0 = 0$, $ak_{(K-1)} = (K - 1)$. $$P(x = k | \theta, \psi) = \frac{exp(ak_{k-1} * (a_1 * \theta_1 + a_2 * \theta_2) + d_{k-1})} {\sum_{k=1}^K exp(ak_{k-1} * (a_1 * \theta_1 + a_2 * \theta_2) + d_{k-1})}$$

For the partial credit model (when itemtype = 'Rasch'; unidimensional only) the above model is further constrained so that $ak = (0,1,\ldots, K-1)$, $a_1 = 1$, and the latent variance of $\theta_1$ is freely estimated. Alternatively, the partial credit model can be obtained by containing all the slope parameters in the gpcms to be equal. More specific scoring function may be included by passing a suitable list or matrices to the gpcm_mats input argument.

In the nominal model this parametrization helps to identify the empirical ordering of the categories by inspecting the $ak$ values. Larger values indicate that the item category is more positively related to the latent trait(s) being measured. For instance, if an item was truly ordinal (such as a Likert scale), and had 4 response categories, we would expect to see $ak_0 < ak_1 < ak_2 < ak_3$ following estimation. If on the other hand $ak_0 > ak_1$ then it would appear that the second category is less related to to the trait than the first, and therefore the second category should be understood as the 'lowest score'.

NOTE: The nominal model can become numerical unstable if poor choices for the high and low values are chosen, resulting in ak values greater than abs(10) or more. It is recommended to choose high and low anchors that cause the estimated parameters to fall between 0 and $K - 1$ either by theoretical means or by re-estimating the model with better values following convergence.

gpcmIRT and rsm

The gpcmIRT model is the classical generalized partial credit model for unidimensional response data. It will obtain the same fit as the gpcm presented above, however the parameterization allows for the Rasch/generalized rating scale model as a special case.

E.g., for a K = 4 category response model,

$$P(x = 0 | \theta, \psi) = exp(0) / G$$ $$P(x = 1 | \theta, \psi) = exp(a(\theta - b1) + c) / G$$ $$P(x = 2 | \theta, \psi) = exp(a(2\theta - b1 - b2) + 2c) / G$$ $$P(x = 3 | \theta, \psi) = exp(a(3\theta - b1 - b2 - b3) + 3c) / G$$ where $$G = exp(0) + exp(a(\theta - b1) + c) + exp(a(2\theta - b1 - b2) + 2c) + exp(a(3\theta - b1 - b2 - b3) + 3c)$$ Here $a$ is the slope parameter, the $b$ parameters are the threshold values for each adjacent category, and $c$ is the so-called difficulty parameter when a rating scale model is fitted (otherwise, $c = 0$ and it drops out of the computations).

The gpcmIRT can be constrained to the partial credit IRT model by either constraining all the slopes to be equal, or setting the slopes to 1 and freeing the latent variance parameter.

Finally, the rsm is a more constrained version of the (generalized) partial credit model where the spacing is equal across item blocks and only adjusted by a single 'difficulty' parameter (c). Note that this is analogous to the relationship between the graded model and the grsm (with an additional constraint regarding the fixed discrimination parameters).

sequential/Tutz

The multidimensional sequential response model has the form $$P(x = k | \theta, \psi) = \prod (1 - F(a_1 \theta_1 + a_2 \theta_2 + d_{sk})) F(a_1 \theta_1 + a_2 \theta_2 + d_{jk})$$ where $F(\cdot)$ is the cumulative logistic function. The Tutz variant of this model (Tutz, 1990) (via itemtype = 'Tutz') assumes that the slope terms are all equal to 1 and the latent variance terms are estimated (i.e., is a Rasch variant).

ideal

The ideal point model has the form, with the upper bound constraint on $d$ set to 0: $$P(x = 1 | \theta, \psi) = exp(-0.5 * (a_1 * \theta_1 + a_2 * \theta_2 + d)^2)$$

partcomp

Partially compensatory models consist of the product of 2PL probability curves. $$P(x = 1 | \theta, \psi) = g + (1 - g) (\frac{1}{1 + exp(-(a_1 * \theta_1 + d_1))}^c_1 * \frac{1}{1 + exp(-(a_2 * \theta_2 + d_2))}^c_2)$$

where $c_1$ and $c_2$ are binary indicator variables reflecting whether the item should include the select compensatory component (1) or not (0). Note that constraining the slopes to be equal across items will reduce the model to Embretson's (Whitely's) multicomponent model (1980).

2-4PLNRM

Nested logistic curves for modeling distractor items. Requires a scoring key. The model is broken into two components for the probability of endorsement. For successful endorsement the probability trace is the 1-4PL model, while for unsuccessful endorsement: $$P(x = 0 | \theta, \psi) = (1 - P_{1-4PL}(x = 1 | \theta, \psi)) * P_{nominal}(x = k | \theta, \psi)$$ which is the product of the complement of the dichotomous trace line with the nominal response model. In the nominal model, the slope parameters defined above are constrained to be 1's, while the last value of the $ak$ is freely estimated.

ggum

The (multidimensional) generalized graded unfolding model is a class of ideal point models useful for ordinal response data. The form is $$P(z=k|\theta,\psi)=\frac{exp\left[\left(z\sqrt{\sum_{d=1}^{D} a_{id}^{2}(\theta_{jd}-b_{id})^{2}}\right)+\sum_{k=0}^{z}\psi_{ik}\right]+ exp\left[\left((M-z)\sqrt{\sum_{d=1}^{D}a_{id}^{2}(\theta_{jd}-b_{id})^{2}}\right)+ \sum_{k=0}^{z}\psi_{ik}\right]}{\sum_{w=0}^{C}\left(exp\left[\left(w \sqrt{\sum_{d=1}^{D}a_{id}^{2}(\theta_{jd}-b_{id})^{2}}\right)+ \sum_{k=0}^{z}\psi_{ik}\right]+exp\left[\left((M-w) \sqrt{\sum_{d=1}^{D}a_{id}^{2}(\theta_{jd}-b_{id})^{2}}\right)+ \sum_{k=0}^{z}\psi_{ik}\right]\right)}$$ where $\theta_{jd}$ is the location of the $j$th individual on the $d$th dimension, $b_{id}$ is the difficulty location of the $i$th item on the $d$th dimension, $a_{id}$ is the discrimination of the $j$th individual on the $d$th dimension (where the discrimination values are constrained to be positive), $\psi_{ik}$ is the $k$th subjective response category threshold for the $i$th item, assumed to be symmetric about the item and constant across dimensions, where $\psi_{ik} = \sum_{d=1}^D a_{id} t_{ik}$ $z = 1,2,\ldots, C$ (where $C$ is the number of categories minus 1), and $M = 2C + 1$.

(g)hcm, (g)alm, (g)sslm, and (g)paralla

Following Luo (2001), this family of response models can be characterized under the same ordinal response functioning structure, differing only in their linking functions ($\psi(x)$). For example, for a two-dimensional model the equation used is $$p_k = \frac{\psi (\rho_k)}{\psi (\rho_k) + \psi (a_1 \theta_1 + a_2 \theta_2 + d)} $$ which is expressed in slope-intercept form to accommodate multidimensionality. The "generalized" versions of this family estimate the slope and rho parameters, which allow each item to differ in the steepness of the unfolding model functions; otherwise, slopes are fixed to the value of 1, though the rho parameters must be set to 0 manually. For ordered polytomous items the response function follows the Guttman-scaling logic $$P_1 = q_1\cdot q_2 \cdot q_3 \cdots q_k$$ $$P_2 = p_1\cdot q_2 \cdot q_3 \cdots q_k$$ $$P_3 = p_1\cdot p_2 \cdot q_3 \cdots q_k$$ $$\cdots$$ $$P_K = p_1\cdot p_2 \cdot p_3 \cdots p_k$$ Note that for estimation purposes the rho parameters are expressed in log units so that they remain positive during estimation. Hence, the parameterization used herein is $$p_k = \frac{\psi (exp(\rho^*_k))}{\psi (exp(\rho^*_k))) + \psi (a_1 \theta_1 + a_2 \theta_2 + d)} $$ where $\rho^*_k$ is in natural log units. Currently supported models in this family are the:

(generalized) hyperbolic cosine model ($\psi (x) = cosh(x)$),
(generalized) absolute logistic model ($\psi (x) = exp(|x|)$),
(generalized) simple squared logistic model ($\psi (x) = exp(x^2)$), and
(generalized) parallellogram analysis model ($\psi (x) = x^2$), respectively.

all of which are available for dichotomous and ordered polytomous response option items.

spline

Spline response models attempt to model the response curves uses non-linear and potentially non-monotonic patterns. The form is $$P(x = 1|\theta, \eta) = \frac{1}{1 + exp(-(\eta_1 * X_1 + \eta_2 * X_2 + \cdots + \eta_n * X_n))}$$ where the $X_n$ are from the spline design matrix $X$ organized from the grid of $\theta$ values. B-splines with a natural or polynomial basis are supported, and the intercept input is set to TRUE by default.

monopoly

Monotone polynomial model for polytomous response data of the form $$P(x = k | \theta, \psi) = \frac{exp(\sum_1^k (m^*(\psi) + \xi_{c-1})} {\sum_1^C exp(\sum_1^K (m^*(\psi) + \xi_{c-1}))}$$ where $m^*(\psi)$ is the monotone polynomial function without the intercept.

HTML help files, exercises, and examples

To access examples, vignettes, and exercise files that have been generated with knitr please visit https://github.com/philchalmers/mirt/wiki.

References

Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561-573.

Andrich, D. (1996). Hyperbolic cosine latent trait models for unfolding direct-responses and pairwise preferences. Applied Psychological Measurement, 20, 269-290.

Andrich, D., and Luo, G. (1993). A hyperbolic cosine latent trait model for unfolding dichotomous single- stimulus responses. Applied Psychological Measurement, 17, 253-276.

Andrich, D. (1988). The application of an unfolding model of the PIRT type to the measurement of attitude. Applied Psychological Measurement, 12, 33-51.

Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.

Bock, R. D., Gibbons, R., & Muraki, E. (1988). Full-Information Item Factor Analysis. Applied Psychological Measurement, 12(3), 261-280.

Bock, R. D. & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179-197.

Cai, L. (2010a). High-Dimensional exploratory item factor analysis by a Metropolis-Hastings Robbins-Monro algorithm. Psychometrika, 75, 33-57.

Cai, L. (2010b). Metropolis-Hastings Robbins-Monro algorithm for confirmatory item factor analysis. Journal of Educational and Behavioral Statistics, 35, 307-335.

Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06

Chalmers, R. P. (2015). Extended Mixed-Effects Item Response Models with the MH-RM Algorithm. Journal of Educational Measurement, 52, 200-222. doi:10.1111/jedm.12072

Chalmers, R. P. (2018). Numerical Approximation of the Observed Information Matrix with Oakes' Identity. British Journal of Mathematical and Statistical Psychology DOI: 10.1111/bmsp.12127

Chalmers, R., P. & Flora, D. (2014). Maximum-likelihood Estimation of Noncompensatory IRT Models with the MH-RM Algorithm. Applied Psychological Measurement, 38, 339-358. doi:10.1177/0146621614520958

Chen, W. H. & Thissen, D. (1997). Local dependence indices for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265-289.

Embretson, S. E. (1984). A general latent trait model for response processes. Psychometrika, 49, 175-186.

Falk, C. F. & Cai, L. (2016). Maximum Marginal Likelihood Estimation of a Monotonic Polynomial Generalized Partial Credit Model with Applications to Multiple Group Analysis. Psychometrika, 81, 434-460.

Fischer, G. H. (1983). Logistic latent trait models with linear constraints. Psychometrika, 48, 3-26.

Hoijtink H. (1990). PARELLA: Measurement of latent traits by proximity items. The Netherlands: University of Groningen.

Lord, F. M. & Novick, M. R. (1968). Statistical theory of mental test scores. Addison-Wesley.

Lucke, J. F. (2015). Unipolar item response models. In S. P. Reise & D. A. Revicki (Eds.), Handbook of item response theory modeling: Applications to typical performance assessment (pp. 272-284). New York, NY: Routledge/Taylor & Francis Group.

Luo G. (2001). A class of probabilistic unfolding models for polytomous responses. Journal of Mathematical Psychology. 45(2):224-248. 10.1006/jmps.2000.1310

Luo G, and Andrich D. (2005). Information functions for the general dichotomous unfolding model. In: Alagumalai S, Curtis D.D., & Hungi N., editor. Applied Rasch Measurement: A Book of Exemplars: Dordrecht, The Netherlands: Springer.

Maydeu-Olivares, A., Hernandez, A. & McDonald, R. P. (2006). A Multidimensional Ideal Point Item Response Theory Model for Binary Data. Multivariate Behavioral Research, 41, 445-471.

Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59-71.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159-176.

Muraki, E. & Carlson, E. B. (1995). Full-information factor analysis for polytomous item responses. Applied Psychological Measurement, 19, 73-90.

Ramsay, J. O. (1975). Solving implicit equations in psychometric data analysis. Psychometrika, 40, 337-360.

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Danish Institute for Educational Research.

Roberts, J. S., Donoghue, J. R., & Laughlin, J. E. (2000). A General Item Response Theory Model for Unfolding Unidimensional Polytomous Responses. Applied Psychological Measurement, 24, 3-32.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monographs, 34.

Shim, H., Bonifay, W., & Wiedermann, W. (2022). Parsimonious asymmetric item response theory modeling with the complementary log-log link. Behavior Research Methods, 55, 200-219.

Suh, Y. & Bolt, D. (2010). Nested logit models for multiple-choice item response data. Psychometrika, 75, 454-473.

Sympson, J. B. (1977). A model for testing with multidimensional items. Proceedings of the 1977 Computerized Adaptive Testing Conference.

Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175-186.

Tutz, G. (1990). Sequential item response models with ordered response. British Journal of Mathematical and Statistical Psychology, 43, 39-55.

Varadhan, R. & Roland, C. (2008). Simple and Globally Convergent Methods for Accelerating the Convergence of Any EM Algorithm. Scandinavian Journal of Statistics, 35, 335-353.

Whitely, S. E. (1980). Multicomponent latent trait models for ability tests. Psychometrika, 45(4), 470-494.

Wood, R., Wilson, D. T., Gibbons, R. D., Schilling, S. G., Muraki, E., & Bock, R. D. (2003). TESTFACT 4 for Windows: Test Scoring, Item Statistics, and Full-information Item Factor Analysis [Computer software]. Lincolnwood, IL: Scientific Software International.

Woods, C. M., and Lin, N. (2009). Item Response Theory With Estimation of the Latent Density Using Davidian Curves. Applied Psychological Measurement,33(2), 102-117.

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples


# load LSAT section 7 data and compute 1 and 2 factor models
data <- expand.table(LSAT7)
itemstats(data)
#> $overall
#>     N mean_total.score sd_total.score ave.r  sd.r alpha SEM.alpha
#>  1000            3.707          1.199 0.143 0.052 0.453     0.886
#> 
#> $itemstats
#>           N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item.1 1000 0.828 0.378   0.530         0.246       0.396
#> Item.2 1000 0.658 0.475   0.600         0.247       0.394
#> Item.3 1000 0.772 0.420   0.611         0.313       0.345
#> Item.4 1000 0.606 0.489   0.592         0.223       0.415
#> Item.5 1000 0.843 0.364   0.461         0.175       0.438
#> 
#> $proportions
#>            0     1
#> Item.1 0.172 0.828
#> Item.2 0.342 0.658
#> Item.3 0.228 0.772
#> Item.4 0.394 0.606
#> Item.5 0.157 0.843
#> 

(mod1 <- mirt(data, 1))
#> 
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2659.430, Max-Change: 0.04181
Iteration: 5, Log-Lik: -2659.241, Max-Change: 0.03417
Iteration: 6, Log-Lik: -2659.113, Max-Change: 0.02911
Iteration: 7, Log-Lik: -2658.812, Max-Change: 0.00456
Iteration: 8, Log-Lik: -2658.809, Max-Change: 0.00363
Iteration: 9, Log-Lik: -2658.808, Max-Change: 0.00273
Iteration: 10, Log-Lik: -2658.806, Max-Change: 0.00144
Iteration: 11, Log-Lik: -2658.806, Max-Change: 0.00118
Iteration: 12, Log-Lik: -2658.806, Max-Change: 0.00101
Iteration: 13, Log-Lik: -2658.805, Max-Change: 0.00042
Iteration: 14, Log-Lik: -2658.805, Max-Change: 0.00025
Iteration: 15, Log-Lik: -2658.805, Max-Change: 0.00026
Iteration: 16, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 17, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 18, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 19, Log-Lik: -2658.805, Max-Change: 0.00019
Iteration: 20, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 21, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 22, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 23, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 24, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 25, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 26, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 27, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 28, Log-Lik: -2658.805, Max-Change: 0.00010
#> 
#> Call:
#> mirt(data = data, model = 1)
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 28 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -2658.805
#> Estimated parameters: 10 
#> AIC = 5337.61
#> BIC = 5386.688; SABIC = 5354.927
#> G2 (21) = 31.7, p = 0.0628
#> RMSEA = 0.023, CFI = NaN, TLI = NaN
coef(mod1)
#> $Item.1
#>        a1     d g u
#> par 0.988 1.856 0 1
#> 
#> $Item.2
#>        a1     d g u
#> par 1.081 0.808 0 1
#> 
#> $Item.3
#>        a1     d g u
#> par 1.706 1.804 0 1
#> 
#> $Item.4
#>        a1     d g u
#> par 0.765 0.486 0 1
#> 
#> $Item.5
#>        a1     d g u
#> par 0.736 1.855 0 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 
summary(mod1)
#>           F1    h2
#> Item.1 0.502 0.252
#> Item.2 0.536 0.287
#> Item.3 0.708 0.501
#> Item.4 0.410 0.168
#> Item.5 0.397 0.157
#> 
#> SS loadings:  1.366 
#> Proportion Var:  0.273 
#> 
#> Factor correlations: 
#> 
#>    F1
#> F1  1
plot(mod1)

plot(mod1, type = 'trace')


# \donttest{
(mod2 <- mirt(data, 1, SE = TRUE)) #standard errors via the Oakes method
#> 
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2659.430, Max-Change: 0.04181
Iteration: 5, Log-Lik: -2659.241, Max-Change: 0.03417
Iteration: 6, Log-Lik: -2659.113, Max-Change: 0.02911
Iteration: 7, Log-Lik: -2658.812, Max-Change: 0.00456
Iteration: 8, Log-Lik: -2658.809, Max-Change: 0.00363
Iteration: 9, Log-Lik: -2658.808, Max-Change: 0.00273
Iteration: 10, Log-Lik: -2658.806, Max-Change: 0.00144
Iteration: 11, Log-Lik: -2658.806, Max-Change: 0.00118
Iteration: 12, Log-Lik: -2658.806, Max-Change: 0.00101
Iteration: 13, Log-Lik: -2658.805, Max-Change: 0.00042
Iteration: 14, Log-Lik: -2658.805, Max-Change: 0.00025
Iteration: 15, Log-Lik: -2658.805, Max-Change: 0.00026
Iteration: 16, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 17, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 18, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 19, Log-Lik: -2658.805, Max-Change: 0.00019
Iteration: 20, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 21, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 22, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 23, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 24, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 25, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 26, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 27, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 28, Log-Lik: -2658.805, Max-Change: 0.00010
#> 
#> Calculating information matrix...
#> 
#> Call:
#> mirt(data = data, model = 1, SE = TRUE)
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 28 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Information matrix estimated with method: Oakes
#> Second-order test: model is a possible local maximum
#> Condition number of information matrix =  30.23088
#> 
#> Log-likelihood = -2658.805
#> Estimated parameters: 10 
#> AIC = 5337.61
#> BIC = 5386.688; SABIC = 5354.927
#> G2 (21) = 31.7, p = 0.0628
#> RMSEA = 0.023, CFI = NaN, TLI = NaN
(mod2 <- mirt(data, 1, SE = TRUE, SE.type = 'SEM')) #standard errors with SEM method
#> 
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2660.367, Max-Change: 0.07790
Iteration: 5, Log-Lik: -2659.792, Max-Change: 0.06078
Iteration: 6, Log-Lik: -2659.461, Max-Change: 0.04772
Iteration: 7, Log-Lik: -2659.253, Max-Change: 0.03901
Iteration: 8, Log-Lik: -2659.116, Max-Change: 0.03157
Iteration: 9, Log-Lik: -2659.025, Max-Change: 0.02660
Iteration: 10, Log-Lik: -2658.960, Max-Change: 0.02198
Iteration: 11, Log-Lik: -2658.916, Max-Change: 0.01850
Iteration: 12, Log-Lik: -2658.885, Max-Change: 0.01570
Iteration: 13, Log-Lik: -2658.863, Max-Change: 0.01339
Iteration: 14, Log-Lik: -2658.847, Max-Change: 0.01200
Iteration: 15, Log-Lik: -2658.835, Max-Change: 0.00987
Iteration: 16, Log-Lik: -2658.827, Max-Change: 0.00854
Iteration: 17, Log-Lik: -2658.821, Max-Change: 0.00667
Iteration: 18, Log-Lik: -2658.817, Max-Change: 0.00586
Iteration: 19, Log-Lik: -2658.814, Max-Change: 0.00509
Iteration: 20, Log-Lik: -2658.812, Max-Change: 0.00442
Iteration: 21, Log-Lik: -2658.810, Max-Change: 0.00348
Iteration: 22, Log-Lik: -2658.809, Max-Change: 0.00322
Iteration: 23, Log-Lik: -2658.808, Max-Change: 0.00256
Iteration: 24, Log-Lik: -2658.807, Max-Change: 0.00252
Iteration: 25, Log-Lik: -2658.807, Max-Change: 0.00199
Iteration: 26, Log-Lik: -2658.807, Max-Change: 0.00197
Iteration: 27, Log-Lik: -2658.806, Max-Change: 0.00157
Iteration: 28, Log-Lik: -2658.806, Max-Change: 0.00145
Iteration: 29, Log-Lik: -2658.806, Max-Change: 0.00128
Iteration: 30, Log-Lik: -2658.806, Max-Change: 0.00116
Iteration: 31, Log-Lik: -2658.806, Max-Change: 0.00099
Iteration: 32, Log-Lik: -2658.805, Max-Change: 0.00043
Iteration: 33, Log-Lik: -2658.805, Max-Change: 0.00039
Iteration: 34, Log-Lik: -2658.805, Max-Change: 0.00038
Iteration: 35, Log-Lik: -2658.805, Max-Change: 0.00035
Iteration: 36, Log-Lik: -2658.805, Max-Change: 0.00034
Iteration: 37, Log-Lik: -2658.805, Max-Change: 0.00032
Iteration: 38, Log-Lik: -2658.805, Max-Change: 0.00030
Iteration: 39, Log-Lik: -2658.805, Max-Change: 0.00142
Iteration: 40, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 41, Log-Lik: -2658.805, Max-Change: 0.00097
Iteration: 42, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 43, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 44, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 45, Log-Lik: -2658.805, Max-Change: 0.00012
Iteration: 46, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 47, Log-Lik: -2658.805, Max-Change: 0.00010
Iteration: 48, Log-Lik: -2658.805, Max-Change: 0.00010
Iteration: 49, Log-Lik: -2658.805, Max-Change: 0.00046
Iteration: 50, Log-Lik: -2658.805, Max-Change: 0.00007
Iteration: 51, Log-Lik: -2658.805, Max-Change: 0.00006
Iteration: 52, Log-Lik: -2658.805, Max-Change: 0.00006
Iteration: 53, Log-Lik: -2658.805, Max-Change: 0.00006
Iteration: 54, Log-Lik: -2658.805, Max-Change: 0.00005
Iteration: 55, Log-Lik: -2658.805, Max-Change: 0.00005
Iteration: 56, Log-Lik: -2658.805, Max-Change: 0.00005
Iteration: 57, Log-Lik: -2658.805, Max-Change: 0.00022
Iteration: 58, Log-Lik: -2658.805, Max-Change: 0.00004
Iteration: 59, Log-Lik: -2658.805, Max-Change: 0.00003
Iteration: 60, Log-Lik: -2658.805, Max-Change: 0.00003
Iteration: 61, Log-Lik: -2658.805, Max-Change: 0.00003
Iteration: 62, Log-Lik: -2658.805, Max-Change: 0.00003
Iteration: 63, Log-Lik: -2658.805, Max-Change: 0.00002
Iteration: 64, Log-Lik: -2658.805, Max-Change: 0.00002
Iteration: 65, Log-Lik: -2658.805, Max-Change: 0.00002
Iteration: 66, Log-Lik: -2658.805, Max-Change: 0.00002
Iteration: 67, Log-Lik: -2658.805, Max-Change: 0.00010
Iteration: 68, Log-Lik: -2658.805, Max-Change: 0.00001
Iteration: 69, Log-Lik: -2658.805, Max-Change: 0.00001
Iteration: 70, Log-Lik: -2658.805, Max-Change: 0.00001
Iteration: 71, Log-Lik: -2658.805, Max-Change: 0.00001
Iteration: 72, Log-Lik: -2658.805, Max-Change: 0.00001
Iteration: 73, Log-Lik: -2658.805, Max-Change: 0.00001
Iteration: 74, Log-Lik: -2658.805, Max-Change: 0.00001
#> 
#> Calculating information matrix...
#> 
#> Call:
#> mirt(data = data, model = 1, SE = TRUE, SE.type = "SEM")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-05 tolerance after 74 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: none 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Information matrix estimated with method: SEM
#> Second-order test: model is a possible local maximum
#> Condition number of information matrix =  30.12751
#> 
#> Log-likelihood = -2658.805
#> Estimated parameters: 10 
#> AIC = 5337.61
#> BIC = 5386.688; SABIC = 5354.927
#> G2 (21) = 31.7, p = 0.0628
#> RMSEA = 0.023, CFI = NaN, TLI = NaN
coef(mod2)
#> $Item.1
#>            a1     d  g  u
#> par     0.988 1.856  0  1
#> CI_2.5  0.639 1.599 NA NA
#> CI_97.5 1.336 2.112 NA NA
#> 
#> $Item.2
#>            a1     d  g  u
#> par     1.081 0.808  0  1
#> CI_2.5  0.755 0.629 NA NA
#> CI_97.5 1.407 0.987 NA NA
#> 
#> $Item.3
#>            a1     d  g  u
#> par     1.707 1.805  0  1
#> CI_2.5  1.086 1.395 NA NA
#> CI_97.5 2.329 2.215 NA NA
#> 
#> $Item.4
#>            a1     d  g  u
#> par     0.765 0.486  0  1
#> CI_2.5  0.500 0.339 NA NA
#> CI_97.5 1.030 0.633 NA NA
#> 
#> $Item.5
#>            a1     d  g  u
#> par     0.736 1.854  0  1
#> CI_2.5  0.437 1.630 NA NA
#> CI_97.5 1.034 2.079 NA NA
#> 
#> $GroupPars
#>         MEAN_1 COV_11
#> par          0      1
#> CI_2.5      NA     NA
#> CI_97.5     NA     NA
#> 
(mod3 <- mirt(data, 1, SE = TRUE, SE.type = 'Richardson')) #with numerical Richardson method
#> 
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2659.430, Max-Change: 0.04181
Iteration: 5, Log-Lik: -2659.241, Max-Change: 0.03417
Iteration: 6, Log-Lik: -2659.113, Max-Change: 0.02911
Iteration: 7, Log-Lik: -2658.812, Max-Change: 0.00456
Iteration: 8, Log-Lik: -2658.809, Max-Change: 0.00363
Iteration: 9, Log-Lik: -2658.808, Max-Change: 0.00273
Iteration: 10, Log-Lik: -2658.806, Max-Change: 0.00144
Iteration: 11, Log-Lik: -2658.806, Max-Change: 0.00118
Iteration: 12, Log-Lik: -2658.806, Max-Change: 0.00101
Iteration: 13, Log-Lik: -2658.805, Max-Change: 0.00042
Iteration: 14, Log-Lik: -2658.805, Max-Change: 0.00025
Iteration: 15, Log-Lik: -2658.805, Max-Change: 0.00026
Iteration: 16, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 17, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 18, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 19, Log-Lik: -2658.805, Max-Change: 0.00019
Iteration: 20, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 21, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 22, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 23, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 24, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 25, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 26, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 27, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 28, Log-Lik: -2658.805, Max-Change: 0.00010
#> 
#> Calculating information matrix...
#> 
#> Call:
#> mirt(data = data, model = 1, SE = TRUE, SE.type = "Richardson")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 28 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Information matrix estimated with method: Richardson
#> Second-order test: model is a possible local maximum
#> Condition number of information matrix =  30.23102
#> 
#> Log-likelihood = -2658.805
#> Estimated parameters: 10 
#> AIC = 5337.61
#> BIC = 5386.688; SABIC = 5354.927
#> G2 (21) = 31.7, p = 0.0628
#> RMSEA = 0.023, CFI = NaN, TLI = NaN
residuals(mod1)
#> LD matrix (lower triangle) and standardized residual correlations (upper triangle)
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.037  -0.020  -0.007   0.001   0.024   0.051 
#> 
#>        Item.1 Item.2 Item.3 Item.4 Item.5
#> Item.1        -0.021 -0.029  0.051  0.049
#> Item.2  0.453         0.033 -0.016 -0.037
#> Item.3  0.854  1.060        -0.012 -0.002
#> Item.4  2.572  0.267  0.153         0.000
#> Item.5  2.389  1.384  0.003  0.000       
plot(mod1) #test score function

plot(mod1, type = 'trace') #trace lines

plot(mod2, type = 'info') #test information

plot(mod2, MI=200) #expected total score with 95% confidence intervals


# estimated 3PL model for item 5 only
(mod1.3PL <- mirt(data, 1, itemtype = c('2PL', '2PL', '2PL', '2PL', '3PL')))
#> 
Iteration: 1, Log-Lik: -2672.054, Max-Change: 0.24544
Iteration: 2, Log-Lik: -2663.819, Max-Change: 0.13674
Iteration: 3, Log-Lik: -2661.462, Max-Change: 0.10270
Iteration: 4, Log-Lik: -2659.415, Max-Change: 0.04081
Iteration: 5, Log-Lik: -2659.229, Max-Change: 0.03430
Iteration: 6, Log-Lik: -2659.100, Max-Change: 0.02876
Iteration: 7, Log-Lik: -2658.813, Max-Change: 0.00848
Iteration: 8, Log-Lik: -2658.806, Max-Change: 0.00676
Iteration: 9, Log-Lik: -2658.803, Max-Change: 0.00505
Iteration: 10, Log-Lik: -2658.799, Max-Change: 0.00378
Iteration: 11, Log-Lik: -2658.798, Max-Change: 0.00349
Iteration: 12, Log-Lik: -2658.797, Max-Change: 0.00290
Iteration: 13, Log-Lik: -2658.796, Max-Change: 0.00239
Iteration: 14, Log-Lik: -2658.795, Max-Change: 0.00188
Iteration: 15, Log-Lik: -2658.795, Max-Change: 0.00163
Iteration: 16, Log-Lik: -2658.794, Max-Change: 0.00101
Iteration: 17, Log-Lik: -2658.794, Max-Change: 0.00057
Iteration: 18, Log-Lik: -2658.794, Max-Change: 0.00043
Iteration: 19, Log-Lik: -2658.794, Max-Change: 0.00044
Iteration: 20, Log-Lik: -2658.794, Max-Change: 0.00040
Iteration: 21, Log-Lik: -2658.794, Max-Change: 0.00038
Iteration: 22, Log-Lik: -2658.794, Max-Change: 0.00031
Iteration: 23, Log-Lik: -2658.794, Max-Change: 0.00032
Iteration: 24, Log-Lik: -2658.794, Max-Change: 0.00029
Iteration: 25, Log-Lik: -2658.794, Max-Change: 0.00028
Iteration: 26, Log-Lik: -2658.794, Max-Change: 0.00025
Iteration: 27, Log-Lik: -2658.794, Max-Change: 0.00024
Iteration: 28, Log-Lik: -2658.794, Max-Change: 0.00021
Iteration: 29, Log-Lik: -2658.794, Max-Change: 0.00021
Iteration: 30, Log-Lik: -2658.794, Max-Change: 0.00019
Iteration: 31, Log-Lik: -2658.794, Max-Change: 0.00018
Iteration: 32, Log-Lik: -2658.794, Max-Change: 0.00016
Iteration: 33, Log-Lik: -2658.794, Max-Change: 0.00016
Iteration: 34, Log-Lik: -2658.794, Max-Change: 0.00014
Iteration: 35, Log-Lik: -2658.794, Max-Change: 0.00014
Iteration: 36, Log-Lik: -2658.794, Max-Change: 0.00063
Iteration: 37, Log-Lik: -2658.794, Max-Change: 0.00039
Iteration: 38, Log-Lik: -2658.794, Max-Change: 0.00028
Iteration: 39, Log-Lik: -2658.794, Max-Change: 0.00019
Iteration: 40, Log-Lik: -2658.794, Max-Change: 0.00039
Iteration: 41, Log-Lik: -2658.794, Max-Change: 0.00028
Iteration: 42, Log-Lik: -2658.794, Max-Change: 0.00020
Iteration: 43, Log-Lik: -2658.794, Max-Change: 0.00008
#> 
#> Call:
#> mirt(data = data, model = 1, itemtype = c("2PL", "2PL", "2PL", 
#>     "2PL", "3PL"))
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 43 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -2658.794
#> Estimated parameters: 11 
#> AIC = 5339.587
#> BIC = 5393.573; SABIC = 5358.636
#> G2 (20) = 31.68, p = 0.0469
#> RMSEA = 0.024, CFI = NaN, TLI = NaN
coef(mod1.3PL)
#> $Item.1
#>        a1     d g u
#> par 0.987 1.855 0 1
#> 
#> $Item.2
#>        a1     d g u
#> par 1.082 0.808 0 1
#> 
#> $Item.3
#>        a1     d g u
#> par 1.706 1.805 0 1
#> 
#> $Item.4
#>        a1     d g u
#> par 0.764 0.486 0 1
#> 
#> $Item.5
#>        a1     d     g u
#> par 0.778 1.643 0.161 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 

# internally g and u pars are stored as logits, so usually a good idea to include normal prior
#  to help stabilize the parameters. For a value around .182 use a mean
#  of -1.5 (since 1 / (1 + exp(-(-1.5))) == .182)
model <- 'F = 1-5
         PRIOR = (5, g, norm, -1.5, 3)'
mod1.3PL.norm <- mirt(data, model, itemtype = c('2PL', '2PL', '2PL', '2PL', '3PL'))
#> 
Iteration: 1, Log-Lik: -2675.466, Max-Change: 0.28143
Iteration: 2, Log-Lik: -2665.875, Max-Change: 0.13677
Iteration: 3, Log-Lik: -2663.485, Max-Change: 0.10277
Iteration: 4, Log-Lik: -2661.455, Max-Change: 0.04260
Iteration: 5, Log-Lik: -2661.259, Max-Change: 0.03555
Iteration: 6, Log-Lik: -2661.125, Max-Change: 0.02961
Iteration: 7, Log-Lik: -2660.829, Max-Change: 0.00835
Iteration: 8, Log-Lik: -2660.823, Max-Change: 0.00636
Iteration: 9, Log-Lik: -2660.819, Max-Change: 0.00487
Iteration: 10, Log-Lik: -2660.814, Max-Change: 0.00390
Iteration: 11, Log-Lik: -2660.812, Max-Change: 0.00267
Iteration: 12, Log-Lik: -2660.811, Max-Change: 0.00234
Iteration: 13, Log-Lik: -2660.810, Max-Change: 0.00169
Iteration: 14, Log-Lik: -2660.810, Max-Change: 0.00153
Iteration: 15, Log-Lik: -2660.810, Max-Change: 0.00138
Iteration: 16, Log-Lik: -2660.809, Max-Change: 0.00070
Iteration: 17, Log-Lik: -2660.809, Max-Change: 0.00051
Iteration: 18, Log-Lik: -2660.809, Max-Change: 0.00023
Iteration: 19, Log-Lik: -2660.809, Max-Change: 0.00025
Iteration: 20, Log-Lik: -2660.809, Max-Change: 0.00025
Iteration: 21, Log-Lik: -2660.809, Max-Change: 0.00113
Iteration: 22, Log-Lik: -2660.809, Max-Change: 0.00016
Iteration: 23, Log-Lik: -2660.809, Max-Change: 0.00068
Iteration: 24, Log-Lik: -2660.809, Max-Change: 0.00014
Iteration: 25, Log-Lik: -2660.809, Max-Change: 0.00012
Iteration: 26, Log-Lik: -2660.809, Max-Change: 0.00009
coef(mod1.3PL.norm)
#> $Item.1
#>        a1     d g u
#> par 0.987 1.855 0 1
#> 
#> $Item.2
#>        a1     d g u
#> par 1.083 0.808 0 1
#> 
#> $Item.3
#>        a1     d g u
#> par 1.706 1.804 0 1
#> 
#> $Item.4
#>        a1     d g u
#> par 0.764 0.486 0 1
#> 
#> $Item.5
#>        a1   d    g u
#> par 0.788 1.6 0.19 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 
#limited information fit statistics
M2(mod1.3PL.norm)
#>             M2 df          p      RMSEA RMSEA_5   RMSEA_95      SRMSR       TLI
#> stats 8.800082  4 0.06629543 0.03465864       0 0.06610847 0.03207363 0.9454563
#>             CFI
#> stats 0.9781825

# unidimensional ideal point model
idealpt <- mirt(data, 1, itemtype = 'ideal')
#> 
Iteration: 1, Log-Lik: -2893.174, Max-Change: 0.26330
Iteration: 2, Log-Lik: -2716.222, Max-Change: 0.06349
Iteration: 3, Log-Lik: -2699.962, Max-Change: 0.04532
Iteration: 4, Log-Lik: -2690.600, Max-Change: 0.03741
Iteration: 5, Log-Lik: -2682.871, Max-Change: 0.04590
Iteration: 6, Log-Lik: -2675.645, Max-Change: 0.04405
Iteration: 7, Log-Lik: -2670.247, Max-Change: 0.03961
Iteration: 8, Log-Lik: -2666.364, Max-Change: 0.03373
Iteration: 9, Log-Lik: -2663.753, Max-Change: 0.02734
Iteration: 10, Log-Lik: -2660.022, Max-Change: 0.00695
Iteration: 11, Log-Lik: -2659.872, Max-Change: 0.00601
Iteration: 12, Log-Lik: -2659.780, Max-Change: 0.00512
Iteration: 13, Log-Lik: -2659.574, Max-Change: 0.00224
Iteration: 14, Log-Lik: -2659.555, Max-Change: 0.00193
Iteration: 15, Log-Lik: -2659.541, Max-Change: 0.00168
Iteration: 16, Log-Lik: -2659.491, Max-Change: 0.00107
Iteration: 17, Log-Lik: -2659.487, Max-Change: 0.00095
Iteration: 18, Log-Lik: -2659.484, Max-Change: 0.00096
Iteration: 19, Log-Lik: -2659.469, Max-Change: 0.00083
Iteration: 20, Log-Lik: -2659.467, Max-Change: 0.00074
Iteration: 21, Log-Lik: -2659.466, Max-Change: 0.00068
Iteration: 22, Log-Lik: -2659.460, Max-Change: 0.00041
Iteration: 23, Log-Lik: -2659.459, Max-Change: 0.00040
Iteration: 24, Log-Lik: -2659.458, Max-Change: 0.00039
Iteration: 25, Log-Lik: -2659.456, Max-Change: 0.00036
Iteration: 26, Log-Lik: -2659.456, Max-Change: 0.00029
Iteration: 27, Log-Lik: -2659.456, Max-Change: 0.00026
Iteration: 28, Log-Lik: -2659.455, Max-Change: 0.00015
Iteration: 29, Log-Lik: -2659.454, Max-Change: 0.00015
Iteration: 30, Log-Lik: -2659.454, Max-Change: 0.00015
Iteration: 31, Log-Lik: -2659.454, Max-Change: 0.00011
Iteration: 32, Log-Lik: -2659.454, Max-Change: 0.00010
Iteration: 33, Log-Lik: -2659.454, Max-Change: 0.00009
plot(idealpt, type = 'trace', facet_items = TRUE)

plot(idealpt, type = 'trace', facet_items = FALSE)


# two factors (exploratory)
mod2 <- mirt(data, 2)
#> 
Iteration: 1, Log-Lik: -2674.021, Max-Change: 0.20368
Iteration: 2, Log-Lik: -2658.770, Max-Change: 0.12215
Iteration: 3, Log-Lik: -2655.896, Max-Change: 0.07195
Iteration: 4, Log-Lik: -2654.819, Max-Change: 0.03372
Iteration: 5, Log-Lik: -2654.630, Max-Change: 0.02022
Iteration: 6, Log-Lik: -2654.553, Max-Change: 0.01189
Iteration: 7, Log-Lik: -2654.493, Max-Change: 0.00789
Iteration: 8, Log-Lik: -2654.475, Max-Change: 0.00763
Iteration: 9, Log-Lik: -2654.460, Max-Change: 0.00744
Iteration: 10, Log-Lik: -2654.379, Max-Change: 0.00615
Iteration: 11, Log-Lik: -2654.368, Max-Change: 0.00600
Iteration: 12, Log-Lik: -2654.357, Max-Change: 0.00589
Iteration: 13, Log-Lik: -2654.297, Max-Change: 0.00550
Iteration: 14, Log-Lik: -2654.288, Max-Change: 0.00545
Iteration: 15, Log-Lik: -2654.280, Max-Change: 0.00540
Iteration: 16, Log-Lik: -2654.232, Max-Change: 0.00523
Iteration: 17, Log-Lik: -2654.224, Max-Change: 0.00515
Iteration: 18, Log-Lik: -2654.218, Max-Change: 0.00509
Iteration: 19, Log-Lik: -2654.179, Max-Change: 0.00497
Iteration: 20, Log-Lik: -2654.173, Max-Change: 0.00492
Iteration: 21, Log-Lik: -2654.167, Max-Change: 0.00487
Iteration: 22, Log-Lik: -2654.134, Max-Change: 0.00466
Iteration: 23, Log-Lik: -2654.130, Max-Change: 0.00460
Iteration: 24, Log-Lik: -2654.125, Max-Change: 0.00455
Iteration: 25, Log-Lik: -2654.097, Max-Change: 0.00437
Iteration: 26, Log-Lik: -2654.093, Max-Change: 0.00434
Iteration: 27, Log-Lik: -2654.089, Max-Change: 0.00431
Iteration: 28, Log-Lik: -2654.066, Max-Change: 0.00420
Iteration: 29, Log-Lik: -2654.062, Max-Change: 0.00416
Iteration: 30, Log-Lik: -2654.058, Max-Change: 0.00413
Iteration: 31, Log-Lik: -2654.038, Max-Change: 0.00404
Iteration: 32, Log-Lik: -2654.034, Max-Change: 0.00401
Iteration: 33, Log-Lik: -2654.031, Max-Change: 0.00399
Iteration: 34, Log-Lik: -2654.012, Max-Change: 0.00391
Iteration: 35, Log-Lik: -2654.009, Max-Change: 0.00389
Iteration: 36, Log-Lik: -2654.006, Max-Change: 0.00387
Iteration: 37, Log-Lik: -2653.989, Max-Change: 0.00373
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Iteration: 351, Log-Lik: -2653.521, Max-Change: 0.00048
Iteration: 352, Log-Lik: -2653.521, Max-Change: 0.00013
Iteration: 353, Log-Lik: -2653.521, Max-Change: 0.00068
Iteration: 354, Log-Lik: -2653.521, Max-Change: 0.00054
Iteration: 355, Log-Lik: -2653.521, Max-Change: 0.00014
Iteration: 356, Log-Lik: -2653.521, Max-Change: 0.00013
Iteration: 357, Log-Lik: -2653.521, Max-Change: 0.00066
Iteration: 358, Log-Lik: -2653.521, Max-Change: 0.00061
Iteration: 359, Log-Lik: -2653.521, Max-Change: 0.00042
Iteration: 360, Log-Lik: -2653.521, Max-Change: 0.00029
Iteration: 361, Log-Lik: -2653.521, Max-Change: 0.00065
Iteration: 362, Log-Lik: -2653.521, Max-Change: 0.00020
Iteration: 363, Log-Lik: -2653.521, Max-Change: 0.00014
Iteration: 364, Log-Lik: -2653.521, Max-Change: 0.00064
Iteration: 365, Log-Lik: -2653.521, Max-Change: 0.00041
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Iteration: 371, Log-Lik: -2653.521, Max-Change: 0.00041
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Iteration: 373, Log-Lik: -2653.521, Max-Change: 0.00062
Iteration: 374, Log-Lik: -2653.521, Max-Change: 0.00020
Iteration: 375, Log-Lik: -2653.521, Max-Change: 0.00013
Iteration: 376, Log-Lik: -2653.521, Max-Change: 0.00061
Iteration: 377, Log-Lik: -2653.521, Max-Change: 0.00039
Iteration: 378, Log-Lik: -2653.521, Max-Change: 0.00027
Iteration: 379, Log-Lik: -2653.521, Max-Change: 0.00061
Iteration: 380, Log-Lik: -2653.520, Max-Change: 0.00018
Iteration: 381, Log-Lik: -2653.520, Max-Change: 0.00013
Iteration: 382, Log-Lik: -2653.520, Max-Change: 0.00060
Iteration: 383, Log-Lik: -2653.520, Max-Change: 0.00042
Iteration: 384, Log-Lik: -2653.520, Max-Change: 0.00029
Iteration: 385, Log-Lik: -2653.520, Max-Change: 0.00060
Iteration: 386, Log-Lik: -2653.520, Max-Change: 0.00027
Iteration: 387, Log-Lik: -2653.520, Max-Change: 0.00019
Iteration: 388, Log-Lik: -2653.520, Max-Change: 0.00059
Iteration: 389, Log-Lik: -2653.520, Max-Change: 0.00012
Iteration: 390, Log-Lik: -2653.520, Max-Change: 0.00058
Iteration: 391, Log-Lik: -2653.520, Max-Change: 0.00069
Iteration: 392, Log-Lik: -2653.520, Max-Change: 0.00048
Iteration: 393, Log-Lik: -2653.520, Max-Change: 0.00034
Iteration: 394, Log-Lik: -2653.520, Max-Change: 0.00058
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Iteration: 396, Log-Lik: -2653.520, Max-Change: 0.00030
Iteration: 397, Log-Lik: -2653.520, Max-Change: 0.00057
Iteration: 398, Log-Lik: -2653.520, Max-Change: 0.00032
Iteration: 399, Log-Lik: -2653.520, Max-Change: 0.00022
Iteration: 400, Log-Lik: -2653.520, Max-Change: 0.00056
Iteration: 401, Log-Lik: -2653.520, Max-Change: 0.00056
Iteration: 402, Log-Lik: -2653.520, Max-Change: 0.00042
Iteration: 403, Log-Lik: -2653.520, Max-Change: 0.00011
Iteration: 404, Log-Lik: -2653.520, Max-Change: 0.00055
Iteration: 405, Log-Lik: -2653.520, Max-Change: 0.00060
Iteration: 406, Log-Lik: -2653.520, Max-Change: 0.00011
Iteration: 407, Log-Lik: -2653.520, Max-Change: 0.00054
Iteration: 408, Log-Lik: -2653.520, Max-Change: 0.00055
Iteration: 409, Log-Lik: -2653.520, Max-Change: 0.00011
Iteration: 410, Log-Lik: -2653.520, Max-Change: 0.00054
Iteration: 411, Log-Lik: -2653.520, Max-Change: 0.00056
Iteration: 412, Log-Lik: -2653.520, Max-Change: 0.00011
Iteration: 413, Log-Lik: -2653.520, Max-Change: 0.00053
Iteration: 414, Log-Lik: -2653.520, Max-Change: 0.00055
Iteration: 415, Log-Lik: -2653.520, Max-Change: 0.00011
Iteration: 416, Log-Lik: -2653.520, Max-Change: 0.00053
Iteration: 417, Log-Lik: -2653.520, Max-Change: 0.00054
Iteration: 418, Log-Lik: -2653.520, Max-Change: 0.00011
Iteration: 419, Log-Lik: -2653.520, Max-Change: 0.00052
Iteration: 420, Log-Lik: -2653.520, Max-Change: 0.00054
Iteration: 421, Log-Lik: -2653.520, Max-Change: 0.00010
Iteration: 422, Log-Lik: -2653.520, Max-Change: 0.00052
Iteration: 423, Log-Lik: -2653.520, Max-Change: 0.00053
Iteration: 424, Log-Lik: -2653.520, Max-Change: 0.00010
Iteration: 425, Log-Lik: -2653.520, Max-Change: 0.00051
Iteration: 426, Log-Lik: -2653.520, Max-Change: 0.00053
Iteration: 427, Log-Lik: -2653.520, Max-Change: 0.00010
Iteration: 428, Log-Lik: -2653.520, Max-Change: 0.00050
Iteration: 429, Log-Lik: -2653.520, Max-Change: 0.00052
Iteration: 430, Log-Lik: -2653.520, Max-Change: 0.00010
Iteration: 431, Log-Lik: -2653.520, Max-Change: 0.00050
Iteration: 432, Log-Lik: -2653.520, Max-Change: 0.00051
Iteration: 433, Log-Lik: -2653.520, Max-Change: 0.00010
Iteration: 434, Log-Lik: -2653.520, Max-Change: 0.00049
Iteration: 435, Log-Lik: -2653.520, Max-Change: 0.00051
Iteration: 436, Log-Lik: -2653.520, Max-Change: 0.00010
coef(mod2)
#> $Item.1
#>         a1   a2     d g u
#> par -2.007 0.87 2.648 0 1
#> 
#> $Item.2
#>         a1     a2     d g u
#> par -0.849 -0.522 0.788 0 1
#> 
#> $Item.3
#>         a1     a2     d g u
#> par -2.153 -1.836 2.483 0 1
#> 
#> $Item.4
#>         a1     a2     d g u
#> par -0.756 -0.028 0.485 0 1
#> 
#> $Item.5
#>         a1 a2     d g u
#> par -0.757  0 1.864 0 1
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 COV_11 COV_21 COV_22
#> par      0      0      1      0      1
#> 
summary(mod2, rotate = 'oblimin') #oblimin rotation
#> 
#> Rotation:  oblimin 
#> 
#> Rotated factor loadings: 
#> 
#>             F1      F2    h2
#> Item.1  0.7944 -0.0111 0.623
#> Item.2  0.0804  0.4630 0.255
#> Item.3 -0.0129  0.8628 0.734
#> Item.4  0.2794  0.1925 0.165
#> Item.5  0.2929  0.1772 0.165
#> 
#> Rotated SS loadings:  0.802 1.027 
#> 
#> Factor correlations: 
#> 
#>       F1 F2
#> F1 1.000   
#> F2 0.463  1
residuals(mod2)
#> LD matrix (lower triangle) and standardized residual correlations (upper triangle)
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.018  -0.001   0.000   0.000   0.002   0.011 
#> 
#>        Item.1 Item.2 Item.3 Item.4 Item.5
#> Item.1        -0.001  0.001  0.002  0.003
#> Item.2  0.001         0.000  0.011 -0.018
#> Item.3  0.001  0.000        -0.002  0.006
#> Item.4  0.002  0.111  0.004        -0.001
#> Item.5  0.008  0.325  0.041  0.001       
plot(mod2)

plot(mod2, rotate = 'oblimin')


anova(mod1, mod2) #compare the two models
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> mod1 5337.610 5354.927 5356.263 5386.688 -2658.805                
#> mod2 5335.039 5359.283 5361.153 5403.748 -2653.520 10.571  4 0.032
scoresfull <- fscores(mod2) #factor scores for each response pattern
head(scoresfull)
#>             F1        F2
#> [1,] -1.700513 -1.711766
#> [2,] -1.700513 -1.711766
#> [3,] -1.700513 -1.711766
#> [4,] -1.700513 -1.711766
#> [5,] -1.700513 -1.711766
#> [6,] -1.700513 -1.711766
scorestable <- fscores(mod2, full.scores = FALSE) #save factor score table
#> 
#> Method:  EAP
#> Rotate:  oblimin
#> 
#> Empirical Reliability:
#> 
#>     F1     F2 
#> 0.2717 0.3565 
head(scorestable)
#>      Item.1 Item.2 Item.3 Item.4 Item.5        F1         F2     SE_F1
#> [1,]      0      0      0      0      0 -1.700513 -1.7117658 0.8233498
#> [2,]      0      0      0      0      1 -1.442213 -1.5315319 0.8291596
#> [3,]      0      0      0      1      0 -1.448997 -1.5246522 0.8289670
#> [4,]      0      0      0      1      1 -1.186286 -1.3432706 0.8376135
#> [5,]      0      0      1      0      0 -1.369479 -0.7080874 0.8344669
#> [6,]      0      0      1      0      1 -1.099364 -0.5102905 0.8455289
#>          SE_F2
#> [1,] 0.7705787
#> [2,] 0.7691522
#> [3,] 0.7691142
#> [4,] 0.7711322
#> [5,] 0.7962954
#> [6,] 0.8101333

# confirmatory (as an example, model is not identified since you need 3 items per factor)
# Two ways to define a confirmatory model: with mirt.model, or with a string

# these model definitions are equivalent
cmodel <- mirt.model('
   F1 = 1,4,5
   F2 = 2,3')
cmodel2 <- 'F1 = 1,4,5
            F2 = 2,3'

cmod <- mirt(data, cmodel)
#> 
Iteration: 1, Log-Lik: -2699.825, Max-Change: 0.12711
Iteration: 2, Log-Lik: -2694.976, Max-Change: 0.11542
Iteration: 3, Log-Lik: -2692.006, Max-Change: 0.09999
Iteration: 4, Log-Lik: -2687.101, Max-Change: 0.03783
Iteration: 5, Log-Lik: -2686.796, Max-Change: 0.02687
Iteration: 6, Log-Lik: -2686.657, Max-Change: 0.01944
Iteration: 7, Log-Lik: -2686.485, Max-Change: 0.01320
Iteration: 8, Log-Lik: -2686.465, Max-Change: 0.01296
Iteration: 9, Log-Lik: -2686.448, Max-Change: 0.01236
Iteration: 10, Log-Lik: -2686.377, Max-Change: 0.00963
Iteration: 11, Log-Lik: -2686.369, Max-Change: 0.00919
Iteration: 12, Log-Lik: -2686.363, Max-Change: 0.00800
Iteration: 13, Log-Lik: -2686.346, Max-Change: 0.00698
Iteration: 14, Log-Lik: -2686.341, Max-Change: 0.00640
Iteration: 15, Log-Lik: -2686.338, Max-Change: 0.00648
Iteration: 16, Log-Lik: -2686.326, Max-Change: 0.00528
Iteration: 17, Log-Lik: -2686.323, Max-Change: 0.00549
Iteration: 18, Log-Lik: -2686.321, Max-Change: 0.00422
Iteration: 19, Log-Lik: -2686.318, Max-Change: 0.00499
Iteration: 20, Log-Lik: -2686.317, Max-Change: 0.00386
Iteration: 21, Log-Lik: -2686.315, Max-Change: 0.00370
Iteration: 22, Log-Lik: -2686.310, Max-Change: 0.00285
Iteration: 23, Log-Lik: -2686.309, Max-Change: 0.00284
Iteration: 24, Log-Lik: -2686.308, Max-Change: 0.00277
Iteration: 25, Log-Lik: -2686.305, Max-Change: 0.00210
Iteration: 26, Log-Lik: -2686.305, Max-Change: 0.00348
Iteration: 27, Log-Lik: -2686.304, Max-Change: 0.00202
Iteration: 28, Log-Lik: -2686.304, Max-Change: 0.00201
Iteration: 29, Log-Lik: -2686.304, Max-Change: 0.00319
Iteration: 30, Log-Lik: -2686.303, Max-Change: 0.00156
Iteration: 31, Log-Lik: -2686.303, Max-Change: 0.00179
Iteration: 32, Log-Lik: -2686.303, Max-Change: 0.00277
Iteration: 33, Log-Lik: -2686.302, Max-Change: 0.00151
Iteration: 34, Log-Lik: -2686.302, Max-Change: 0.00150
Iteration: 35, Log-Lik: -2686.302, Max-Change: 0.00145
Iteration: 36, Log-Lik: -2686.302, Max-Change: 0.00137
Iteration: 37, Log-Lik: -2686.302, Max-Change: 0.00136
Iteration: 38, Log-Lik: -2686.301, Max-Change: 0.00298
Iteration: 39, Log-Lik: -2686.301, Max-Change: 0.00136
Iteration: 40, Log-Lik: -2686.301, Max-Change: 0.00233
Iteration: 41, Log-Lik: -2686.301, Max-Change: 0.00117
Iteration: 42, Log-Lik: -2686.301, Max-Change: 0.00112
Iteration: 43, Log-Lik: -2686.300, Max-Change: 0.00228
Iteration: 44, Log-Lik: -2686.300, Max-Change: 0.00054
Iteration: 45, Log-Lik: -2686.300, Max-Change: 0.00037
Iteration: 46, Log-Lik: -2686.300, Max-Change: 0.00042
Iteration: 47, Log-Lik: -2686.300, Max-Change: 0.00039
Iteration: 48, Log-Lik: -2686.300, Max-Change: 0.00041
Iteration: 49, Log-Lik: -2686.300, Max-Change: 0.00036
Iteration: 50, Log-Lik: -2686.300, Max-Change: 0.00041
Iteration: 51, Log-Lik: -2686.300, Max-Change: 0.00185
Iteration: 52, Log-Lik: -2686.299, Max-Change: 0.00207
Iteration: 53, Log-Lik: -2686.299, Max-Change: 0.00027
Iteration: 54, Log-Lik: -2686.299, Max-Change: 0.00031
Iteration: 55, Log-Lik: -2686.299, Max-Change: 0.00031
Iteration: 56, Log-Lik: -2686.299, Max-Change: 0.00033
Iteration: 57, Log-Lik: -2686.299, Max-Change: 0.00159
Iteration: 58, Log-Lik: -2686.299, Max-Change: 0.00035
Iteration: 59, Log-Lik: -2686.299, Max-Change: 0.00028
Iteration: 60, Log-Lik: -2686.299, Max-Change: 0.00032
Iteration: 61, Log-Lik: -2686.299, Max-Change: 0.00143
Iteration: 62, Log-Lik: -2686.299, Max-Change: 0.00080
Iteration: 63, Log-Lik: -2686.299, Max-Change: 0.00053
Iteration: 64, Log-Lik: -2686.299, Max-Change: 0.00026
Iteration: 65, Log-Lik: -2686.299, Max-Change: 0.00029
Iteration: 66, Log-Lik: -2686.299, Max-Change: 0.00134
Iteration: 67, Log-Lik: -2686.299, Max-Change: 0.00046
Iteration: 68, Log-Lik: -2686.299, Max-Change: 0.00031
Iteration: 69, Log-Lik: -2686.299, Max-Change: 0.00028
Iteration: 70, Log-Lik: -2686.299, Max-Change: 0.00123
Iteration: 71, Log-Lik: -2686.299, Max-Change: 0.00028
Iteration: 72, Log-Lik: -2686.299, Max-Change: 0.00021
Iteration: 73, Log-Lik: -2686.299, Max-Change: 0.00120
Iteration: 74, Log-Lik: -2686.299, Max-Change: 0.00027
Iteration: 75, Log-Lik: -2686.299, Max-Change: 0.00024
Iteration: 76, Log-Lik: -2686.299, Max-Change: 0.00108
Iteration: 77, Log-Lik: -2686.299, Max-Change: 0.00025
Iteration: 78, Log-Lik: -2686.299, Max-Change: 0.00019
Iteration: 79, Log-Lik: -2686.299, Max-Change: 0.00105
Iteration: 80, Log-Lik: -2686.299, Max-Change: 0.00022
Iteration: 81, Log-Lik: -2686.299, Max-Change: 0.00021
Iteration: 82, Log-Lik: -2686.298, Max-Change: 0.00095
Iteration: 83, Log-Lik: -2686.298, Max-Change: 0.00028
Iteration: 84, Log-Lik: -2686.298, Max-Change: 0.00019
Iteration: 85, Log-Lik: -2686.298, Max-Change: 0.00091
Iteration: 86, Log-Lik: -2686.298, Max-Change: 0.00086
Iteration: 87, Log-Lik: -2686.298, Max-Change: 0.00019
Iteration: 88, Log-Lik: -2686.298, Max-Change: 0.00018
Iteration: 89, Log-Lik: -2686.298, Max-Change: 0.00078
Iteration: 90, Log-Lik: -2686.298, Max-Change: 0.00041
Iteration: 91, Log-Lik: -2686.298, Max-Change: 0.00017
Iteration: 92, Log-Lik: -2686.298, Max-Change: 0.00014
Iteration: 93, Log-Lik: -2686.298, Max-Change: 0.00016
Iteration: 94, Log-Lik: -2686.298, Max-Change: 0.00014
Iteration: 95, Log-Lik: -2686.298, Max-Change: 0.00016
Iteration: 96, Log-Lik: -2686.298, Max-Change: 0.00072
Iteration: 97, Log-Lik: -2686.298, Max-Change: 0.00028
Iteration: 98, Log-Lik: -2686.298, Max-Change: 0.00019
Iteration: 99, Log-Lik: -2686.298, Max-Change: 0.00015
Iteration: 100, Log-Lik: -2686.298, Max-Change: 0.00067
Iteration: 101, Log-Lik: -2686.298, Max-Change: 0.00014
Iteration: 102, Log-Lik: -2686.298, Max-Change: 0.00012
Iteration: 103, Log-Lik: -2686.298, Max-Change: 0.00013
Iteration: 104, Log-Lik: -2686.298, Max-Change: 0.00061
Iteration: 105, Log-Lik: -2686.298, Max-Change: 0.00023
Iteration: 106, Log-Lik: -2686.298, Max-Change: 0.00013
Iteration: 107, Log-Lik: -2686.298, Max-Change: 0.00011
Iteration: 108, Log-Lik: -2686.298, Max-Change: 0.00012
Iteration: 109, Log-Lik: -2686.298, Max-Change: 0.00011
Iteration: 110, Log-Lik: -2686.298, Max-Change: 0.00012
Iteration: 111, Log-Lik: -2686.298, Max-Change: 0.00056
Iteration: 112, Log-Lik: -2686.298, Max-Change: 0.00022
Iteration: 113, Log-Lik: -2686.298, Max-Change: 0.00015
Iteration: 114, Log-Lik: -2686.298, Max-Change: 0.00012
Iteration: 115, Log-Lik: -2686.298, Max-Change: 0.00052
Iteration: 116, Log-Lik: -2686.298, Max-Change: 0.00011
Iteration: 117, Log-Lik: -2686.298, Max-Change: 0.00048
Iteration: 118, Log-Lik: -2686.298, Max-Change: 0.00032
Iteration: 119, Log-Lik: -2686.298, Max-Change: 0.00022
Iteration: 120, Log-Lik: -2686.298, Max-Change: 0.00014
Iteration: 121, Log-Lik: -2686.298, Max-Change: 0.00048
Iteration: 122, Log-Lik: -2686.298, Max-Change: 0.00016
Iteration: 123, Log-Lik: -2686.298, Max-Change: 0.00010
Iteration: 124, Log-Lik: -2686.298, Max-Change: 0.00044
Iteration: 125, Log-Lik: -2686.298, Max-Change: 0.00008
# cmod <- mirt(data, cmodel2) # same as above
coef(cmod)
#> $Item.1
#>        a1 a2     d g u
#> par 1.792  0 2.358 0 1
#> 
#> $Item.2
#>     a1    a2   d g u
#> par  0 1.427 0.9 0 1
#> 
#> $Item.3
#>     a1    a2     d g u
#> par  0 1.559 1.725 0 1
#> 
#> $Item.4
#>        a1 a2     d g u
#> par 0.743  0 0.483 0 1
#> 
#> $Item.5
#>        a1 a2     d g u
#> par 0.763  0 1.867 0 1
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 COV_11 COV_21 COV_22
#> par      0      0      1      0      1
#> 
anova(cmod, mod2)
#>           AIC    SABIC       HQ      BIC    logLik     X2 df p
#> cmod 5392.596 5409.913 5411.249 5441.674 -2686.298            
#> mod2 5335.039 5359.283 5361.153 5403.748 -2653.520 65.557  4 0
# check if identified by computing information matrix
(cmod <- mirt(data, cmodel, SE = TRUE))
#> 
Iteration: 1, Log-Lik: -2699.825, Max-Change: 0.12711
Iteration: 2, Log-Lik: -2694.976, Max-Change: 0.11542
Iteration: 3, Log-Lik: -2692.006, Max-Change: 0.09999
Iteration: 4, Log-Lik: -2687.101, Max-Change: 0.03783
Iteration: 5, Log-Lik: -2686.796, Max-Change: 0.02687
Iteration: 6, Log-Lik: -2686.657, Max-Change: 0.01944
Iteration: 7, Log-Lik: -2686.485, Max-Change: 0.01320
Iteration: 8, Log-Lik: -2686.465, Max-Change: 0.01296
Iteration: 9, Log-Lik: -2686.448, Max-Change: 0.01236
Iteration: 10, Log-Lik: -2686.377, Max-Change: 0.00963
Iteration: 11, Log-Lik: -2686.369, Max-Change: 0.00919
Iteration: 12, Log-Lik: -2686.363, Max-Change: 0.00800
Iteration: 13, Log-Lik: -2686.346, Max-Change: 0.00698
Iteration: 14, Log-Lik: -2686.341, Max-Change: 0.00640
Iteration: 15, Log-Lik: -2686.338, Max-Change: 0.00648
Iteration: 16, Log-Lik: -2686.326, Max-Change: 0.00528
Iteration: 17, Log-Lik: -2686.323, Max-Change: 0.00549
Iteration: 18, Log-Lik: -2686.321, Max-Change: 0.00422
Iteration: 19, Log-Lik: -2686.318, Max-Change: 0.00499
Iteration: 20, Log-Lik: -2686.317, Max-Change: 0.00386
Iteration: 21, Log-Lik: -2686.315, Max-Change: 0.00370
Iteration: 22, Log-Lik: -2686.310, Max-Change: 0.00285
Iteration: 23, Log-Lik: -2686.309, Max-Change: 0.00284
Iteration: 24, Log-Lik: -2686.308, Max-Change: 0.00277
Iteration: 25, Log-Lik: -2686.305, Max-Change: 0.00210
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Iteration: 123, Log-Lik: -2686.298, Max-Change: 0.00010
Iteration: 124, Log-Lik: -2686.298, Max-Change: 0.00044
Iteration: 125, Log-Lik: -2686.298, Max-Change: 0.00008
#> 
#> Calculating information matrix...
#> Warning: Could not invert information matrix; model may not be (empirically) identified.
#> 
#> Call:
#> mirt(data = data, model = cmodel, SE = TRUE)
#> 
#> Full-information item factor analysis with 2 factor(s).
#> Converged within 1e-04 tolerance after 125 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 31
#> Latent density type: Gaussian 
#> 
#> Information matrix estimated with method: Oakes
#> Second-order test: model is not a maximum or the information matrix is too inaccurate
#> 
#> Log-likelihood = -2686.298
#> Estimated parameters: 10 
#> AIC = 5392.596
#> BIC = 5441.674; SABIC = 5409.913
#> G2 (21) = 86.69, p = 0
#> RMSEA = 0.056, CFI = NaN, TLI = NaN

###########
# data from the 'ltm' package in numeric format
itemstats(Science)
#> $overall
#>    N mean_total.score sd_total.score ave.r  sd.r alpha SEM.alpha
#>  392           11.668          2.003 0.275 0.098 0.598      1.27
#> 
#> $itemstats
#>           N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Comfort 392 3.120 0.588   0.596         0.352       0.552
#> Work    392 2.722 0.807   0.666         0.332       0.567
#> Future  392 2.990 0.757   0.748         0.488       0.437
#> Benefit 392 2.837 0.802   0.684         0.363       0.541
#> 
#> $proportions
#>             1     2     3     4
#> Comfort 0.013 0.082 0.679 0.227
#> Work    0.084 0.250 0.526 0.140
#> Future  0.036 0.184 0.536 0.245
#> Benefit 0.054 0.255 0.492 0.199
#> 

pmod1 <- mirt(Science, 1)
#> 
Iteration: 1, Log-Lik: -1629.361, Max-Change: 0.50660
Iteration: 2, Log-Lik: -1617.374, Max-Change: 0.25442
Iteration: 3, Log-Lik: -1612.894, Max-Change: 0.16991
Iteration: 4, Log-Lik: -1610.306, Max-Change: 0.10461
Iteration: 5, Log-Lik: -1609.814, Max-Change: 0.09162
Iteration: 6, Log-Lik: -1609.534, Max-Change: 0.07363
Iteration: 7, Log-Lik: -1609.030, Max-Change: 0.03677
Iteration: 8, Log-Lik: -1608.988, Max-Change: 0.03200
Iteration: 9, Log-Lik: -1608.958, Max-Change: 0.02754
Iteration: 10, Log-Lik: -1608.878, Max-Change: 0.01443
Iteration: 11, Log-Lik: -1608.875, Max-Change: 0.00847
Iteration: 12, Log-Lik: -1608.873, Max-Change: 0.00515
Iteration: 13, Log-Lik: -1608.872, Max-Change: 0.00550
Iteration: 14, Log-Lik: -1608.872, Max-Change: 0.00318
Iteration: 15, Log-Lik: -1608.871, Max-Change: 0.00462
Iteration: 16, Log-Lik: -1608.871, Max-Change: 0.00277
Iteration: 17, Log-Lik: -1608.870, Max-Change: 0.00145
Iteration: 18, Log-Lik: -1608.870, Max-Change: 0.00175
Iteration: 19, Log-Lik: -1608.870, Max-Change: 0.00126
Iteration: 20, Log-Lik: -1608.870, Max-Change: 0.00025
Iteration: 21, Log-Lik: -1608.870, Max-Change: 0.00285
Iteration: 22, Log-Lik: -1608.870, Max-Change: 0.00108
Iteration: 23, Log-Lik: -1608.870, Max-Change: 0.00022
Iteration: 24, Log-Lik: -1608.870, Max-Change: 0.00059
Iteration: 25, Log-Lik: -1608.870, Max-Change: 0.00014
Iteration: 26, Log-Lik: -1608.870, Max-Change: 0.00068
Iteration: 27, Log-Lik: -1608.870, Max-Change: 0.00065
Iteration: 28, Log-Lik: -1608.870, Max-Change: 0.00019
Iteration: 29, Log-Lik: -1608.870, Max-Change: 0.00061
Iteration: 30, Log-Lik: -1608.870, Max-Change: 0.00012
Iteration: 31, Log-Lik: -1608.870, Max-Change: 0.00012
Iteration: 32, Log-Lik: -1608.870, Max-Change: 0.00058
Iteration: 33, Log-Lik: -1608.870, Max-Change: 0.00055
Iteration: 34, Log-Lik: -1608.870, Max-Change: 0.00015
Iteration: 35, Log-Lik: -1608.870, Max-Change: 0.00052
Iteration: 36, Log-Lik: -1608.870, Max-Change: 0.00010
plot(pmod1)

plot(pmod1, type = 'trace')

plot(pmod1, type = 'itemscore')

summary(pmod1)
#>            F1    h2
#> Comfort 0.522 0.273
#> Work    0.584 0.342
#> Future  0.803 0.645
#> Benefit 0.541 0.293
#> 
#> SS loadings:  1.552 
#> Proportion Var:  0.388 
#> 
#> Factor correlations: 
#> 
#>    F1
#> F1  1

# Constrain all slopes to be equal with the constrain = list() input or mirt.model() syntax
# first obtain parameter index
values <- mirt(Science,1, pars = 'values')
values #note that slopes are numbered 1,5,9,13, or index with values$parnum[values$name == 'a1']
#>    group    item     class   name parnum  value lbound ubound   est const
#> 1    all Comfort    graded     a1      1  0.851   -Inf    Inf  TRUE  none
#> 2    all Comfort    graded     d1      2  4.390   -Inf    Inf  TRUE  none
#> 3    all Comfort    graded     d2      3  2.583   -Inf    Inf  TRUE  none
#> 4    all Comfort    graded     d3      4 -1.471   -Inf    Inf  TRUE  none
#> 5    all    Work    graded     a1      5  0.851   -Inf    Inf  TRUE  none
#> 6    all    Work    graded     d1      6  2.707   -Inf    Inf  TRUE  none
#> 7    all    Work    graded     d2      7  0.842   -Inf    Inf  TRUE  none
#> 8    all    Work    graded     d3      8 -2.120   -Inf    Inf  TRUE  none
#> 9    all  Future    graded     a1      9  0.851   -Inf    Inf  TRUE  none
#> 10   all  Future    graded     d1     10  3.543   -Inf    Inf  TRUE  none
#> 11   all  Future    graded     d2     11  1.522   -Inf    Inf  TRUE  none
#> 12   all  Future    graded     d3     12 -1.357   -Inf    Inf  TRUE  none
#> 13   all Benefit    graded     a1     13  0.851   -Inf    Inf  TRUE  none
#> 14   all Benefit    graded     d1     14  3.166   -Inf    Inf  TRUE  none
#> 15   all Benefit    graded     d2     15  0.982   -Inf    Inf  TRUE  none
#> 16   all Benefit    graded     d3     16 -1.661   -Inf    Inf  TRUE  none
#> 17   all   GROUP GroupPars MEAN_1     17  0.000   -Inf    Inf FALSE  none
#> 18   all   GROUP GroupPars COV_11     18  1.000      0    Inf FALSE  none
#>    nconst prior.type prior_1 prior_2
#> 1    none       none     NaN     NaN
#> 2    none       none     NaN     NaN
#> 3    none       none     NaN     NaN
#> 4    none       none     NaN     NaN
#> 5    none       none     NaN     NaN
#> 6    none       none     NaN     NaN
#> 7    none       none     NaN     NaN
#> 8    none       none     NaN     NaN
#> 9    none       none     NaN     NaN
#> 10   none       none     NaN     NaN
#> 11   none       none     NaN     NaN
#> 12   none       none     NaN     NaN
#> 13   none       none     NaN     NaN
#> 14   none       none     NaN     NaN
#> 15   none       none     NaN     NaN
#> 16   none       none     NaN     NaN
#> 17   none       none     NaN     NaN
#> 18   none       none     NaN     NaN
(pmod1_equalslopes <- mirt(Science, 1, constrain = list(c(1,5,9,13))))
#> 
Iteration: 1, Log-Lik: -1629.361, Max-Change: 0.52419
Iteration: 2, Log-Lik: -1618.990, Max-Change: 0.13337
Iteration: 3, Log-Lik: -1615.588, Max-Change: 0.07100
Iteration: 4, Log-Lik: -1614.373, Max-Change: 0.03752
Iteration: 5, Log-Lik: -1614.057, Max-Change: 0.02162
Iteration: 6, Log-Lik: -1613.952, Max-Change: 0.01259
Iteration: 7, Log-Lik: -1613.909, Max-Change: 0.00555
Iteration: 8, Log-Lik: -1613.902, Max-Change: 0.00334
Iteration: 9, Log-Lik: -1613.900, Max-Change: 0.00173
Iteration: 10, Log-Lik: -1613.900, Max-Change: 0.00133
Iteration: 11, Log-Lik: -1613.899, Max-Change: 0.00051
Iteration: 12, Log-Lik: -1613.899, Max-Change: 0.00022
Iteration: 13, Log-Lik: -1613.899, Max-Change: 0.00018
Iteration: 14, Log-Lik: -1613.899, Max-Change: 0.00013
Iteration: 15, Log-Lik: -1613.899, Max-Change: 0.00009
#> 
#> Call:
#> mirt(data = Science, model = 1, constrain = list(c(1, 5, 9, 13)))
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 15 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -1613.899
#> Estimated parameters: 13 
#> AIC = 3253.798
#> BIC = 3305.425; SABIC = 3264.176
#> G2 (242) = 223.62, p = 0.7959
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(pmod1_equalslopes)
#> $Comfort
#>        a1    d1    d2     d3
#> par 1.321 5.165 2.844 -1.587
#> 
#> $Work
#>        a1    d1    d2     d3
#> par 1.321 2.992 0.934 -2.319
#> 
#> $Future
#>        a1    d1    d2     d3
#> par 1.321 4.067 1.662 -1.488
#> 
#> $Benefit
#>        a1   d1    d2     d3
#> par 1.321 3.55 1.057 -1.806
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 

# using mirt.model syntax, constrain all item slopes to be equal
model <- 'F = 1-4
          CONSTRAIN = (1-4, a1)'
(pmod1_equalslopes <- mirt(Science, model))
#> 
Iteration: 1, Log-Lik: -1629.361, Max-Change: 0.52419
Iteration: 2, Log-Lik: -1618.990, Max-Change: 0.13337
Iteration: 3, Log-Lik: -1615.588, Max-Change: 0.07100
Iteration: 4, Log-Lik: -1614.373, Max-Change: 0.03752
Iteration: 5, Log-Lik: -1614.057, Max-Change: 0.02162
Iteration: 6, Log-Lik: -1613.952, Max-Change: 0.01259
Iteration: 7, Log-Lik: -1613.909, Max-Change: 0.00555
Iteration: 8, Log-Lik: -1613.902, Max-Change: 0.00334
Iteration: 9, Log-Lik: -1613.900, Max-Change: 0.00173
Iteration: 10, Log-Lik: -1613.900, Max-Change: 0.00133
Iteration: 11, Log-Lik: -1613.899, Max-Change: 0.00051
Iteration: 12, Log-Lik: -1613.899, Max-Change: 0.00022
Iteration: 13, Log-Lik: -1613.899, Max-Change: 0.00018
Iteration: 14, Log-Lik: -1613.899, Max-Change: 0.00013
Iteration: 15, Log-Lik: -1613.899, Max-Change: 0.00009
#> 
#> Call:
#> mirt(data = Science, model = model)
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 15 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -1613.899
#> Estimated parameters: 13 
#> AIC = 3253.798
#> BIC = 3305.425; SABIC = 3264.176
#> G2 (242) = 223.62, p = 0.7959
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(pmod1_equalslopes)
#> $Comfort
#>        a1    d1    d2     d3
#> par 1.321 5.165 2.844 -1.587
#> 
#> $Work
#>        a1    d1    d2     d3
#> par 1.321 2.992 0.934 -2.319
#> 
#> $Future
#>        a1    d1    d2     d3
#> par 1.321 4.067 1.662 -1.488
#> 
#> $Benefit
#>        a1   d1    d2     d3
#> par 1.321 3.55 1.057 -1.806
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 

coef(pmod1_equalslopes)
#> $Comfort
#>        a1    d1    d2     d3
#> par 1.321 5.165 2.844 -1.587
#> 
#> $Work
#>        a1    d1    d2     d3
#> par 1.321 2.992 0.934 -2.319
#> 
#> $Future
#>        a1    d1    d2     d3
#> par 1.321 4.067 1.662 -1.488
#> 
#> $Benefit
#>        a1   d1    d2     d3
#> par 1.321 3.55 1.057 -1.806
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 
anova(pmod1_equalslopes, pmod1) #significantly worse fit with almost all criteria
#>                        AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> pmod1_equalslopes 3253.798 3264.176 3274.259 3305.425 -1613.899                
#> pmod1             3249.739 3262.512 3274.922 3313.279 -1608.870 10.059  3 0.018

pmod2 <- mirt(Science, 2)
#> 
Iteration: 1, Log-Lik: -1634.562, Max-Change: 0.49365
Iteration: 2, Log-Lik: -1608.496, Max-Change: 0.17069
Iteration: 3, Log-Lik: -1604.784, Max-Change: 0.08523
Iteration: 4, Log-Lik: -1604.195, Max-Change: 0.05345
Iteration: 5, Log-Lik: -1603.777, Max-Change: 0.03491
Iteration: 6, Log-Lik: -1603.581, Max-Change: 0.02948
Iteration: 7, Log-Lik: -1603.263, Max-Change: 0.02614
Iteration: 8, Log-Lik: -1603.177, Max-Change: 0.02281
Iteration: 9, Log-Lik: -1603.099, Max-Change: 0.02311
Iteration: 10, Log-Lik: -1602.717, Max-Change: 0.01961
Iteration: 11, Log-Lik: -1602.673, Max-Change: 0.01556
Iteration: 12, Log-Lik: -1602.635, Max-Change: 0.01694
Iteration: 13, Log-Lik: -1602.443, Max-Change: 0.01534
Iteration: 14, Log-Lik: -1602.421, Max-Change: 0.01311
Iteration: 15, Log-Lik: -1602.401, Max-Change: 0.01384
Iteration: 16, Log-Lik: -1602.303, Max-Change: 0.00987
Iteration: 17, Log-Lik: -1602.288, Max-Change: 0.01006
Iteration: 18, Log-Lik: -1602.275, Max-Change: 0.00939
Iteration: 19, Log-Lik: -1602.206, Max-Change: 0.00972
Iteration: 20, Log-Lik: -1602.198, Max-Change: 0.01016
Iteration: 21, Log-Lik: -1602.189, Max-Change: 0.00670
Iteration: 22, Log-Lik: -1602.165, Max-Change: 0.00679
Iteration: 23, Log-Lik: -1602.158, Max-Change: 0.00694
Iteration: 24, Log-Lik: -1602.152, Max-Change: 0.00555
Iteration: 25, Log-Lik: -1602.116, Max-Change: 0.00689
Iteration: 26, Log-Lik: -1602.112, Max-Change: 0.00465
Iteration: 27, Log-Lik: -1602.107, Max-Change: 0.00478
Iteration: 28, Log-Lik: -1602.082, Max-Change: 0.00389
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Iteration: 30, Log-Lik: -1602.074, Max-Change: 0.00380
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summary(pmod2)
#> 
#> Rotation:  oblimin 
#> 
#> Rotated factor loadings: 
#> 
#>              F1      F2    h2
#> Comfort  0.6016  0.0312 0.382
#> Work    -0.0573  0.7971 0.592
#> Future   0.3302  0.5153 0.548
#> Benefit  0.7231 -0.0239 0.506
#> 
#> Rotated SS loadings:  0.997 0.902 
#> 
#> Factor correlations: 
#> 
#>       F1 F2
#> F1 1.000   
#> F2 0.511  1
plot(pmod2, rotate = 'oblimin')

itemplot(pmod2, 1, rotate = 'oblimin')

anova(pmod1, pmod2)
#>            AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> pmod1 3249.739 3262.512 3274.922 3313.279 -1608.870                
#> pmod2 3241.938 3257.106 3271.843 3317.392 -1601.969 13.801  3 0.003

# unidimensional fit with a generalized partial credit and nominal model
(gpcmod <- mirt(Science, 1, 'gpcm'))
#> 
Iteration: 1, Log-Lik: -1689.735, Max-Change: 1.30401
Iteration: 2, Log-Lik: -1618.015, Max-Change: 0.28548
Iteration: 3, Log-Lik: -1615.635, Max-Change: 0.27916
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Iteration: 11, Log-Lik: -1612.724, Max-Change: 0.02653
Iteration: 12, Log-Lik: -1612.718, Max-Change: 0.03424
Iteration: 13, Log-Lik: -1612.700, Max-Change: 0.00646
Iteration: 14, Log-Lik: -1612.697, Max-Change: 0.01774
Iteration: 15, Log-Lik: -1612.695, Max-Change: 0.01250
Iteration: 16, Log-Lik: -1612.692, Max-Change: 0.01716
Iteration: 17, Log-Lik: -1612.690, Max-Change: 0.01090
Iteration: 18, Log-Lik: -1612.689, Max-Change: 0.01170
Iteration: 19, Log-Lik: -1612.687, Max-Change: 0.00599
Iteration: 20, Log-Lik: -1612.687, Max-Change: 0.00160
Iteration: 21, Log-Lik: -1612.687, Max-Change: 0.00208
Iteration: 22, Log-Lik: -1612.687, Max-Change: 0.02543
Iteration: 23, Log-Lik: -1612.685, Max-Change: 0.00161
Iteration: 24, Log-Lik: -1612.685, Max-Change: 0.00099
Iteration: 25, Log-Lik: -1612.685, Max-Change: 0.00044
Iteration: 26, Log-Lik: -1612.685, Max-Change: 0.00027
Iteration: 27, Log-Lik: -1612.685, Max-Change: 0.00102
Iteration: 28, Log-Lik: -1612.685, Max-Change: 0.00049
Iteration: 29, Log-Lik: -1612.685, Max-Change: 0.00020
Iteration: 30, Log-Lik: -1612.685, Max-Change: 0.00019
Iteration: 31, Log-Lik: -1612.685, Max-Change: 0.01678
Iteration: 32, Log-Lik: -1612.684, Max-Change: 0.00133
Iteration: 33, Log-Lik: -1612.684, Max-Change: 0.00078
Iteration: 34, Log-Lik: -1612.684, Max-Change: 0.00030
Iteration: 35, Log-Lik: -1612.684, Max-Change: 0.00090
Iteration: 36, Log-Lik: -1612.684, Max-Change: 0.00027
Iteration: 37, Log-Lik: -1612.684, Max-Change: 0.00014
Iteration: 38, Log-Lik: -1612.684, Max-Change: 0.00083
Iteration: 39, Log-Lik: -1612.684, Max-Change: 0.00024
Iteration: 40, Log-Lik: -1612.684, Max-Change: 0.00012
Iteration: 41, Log-Lik: -1612.684, Max-Change: 0.01511
Iteration: 42, Log-Lik: -1612.683, Max-Change: 0.00108
Iteration: 43, Log-Lik: -1612.683, Max-Change: 0.00105
Iteration: 44, Log-Lik: -1612.683, Max-Change: 0.00038
Iteration: 45, Log-Lik: -1612.683, Max-Change: 0.00111
Iteration: 46, Log-Lik: -1612.683, Max-Change: 0.00022
Iteration: 47, Log-Lik: -1612.683, Max-Change: 0.00015
Iteration: 48, Log-Lik: -1612.683, Max-Change: 0.00063
Iteration: 49, Log-Lik: -1612.683, Max-Change: 0.00021
Iteration: 50, Log-Lik: -1612.683, Max-Change: 0.00009
#> 
#> Call:
#> mirt(data = Science, model = 1, itemtype = "gpcm")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 50 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -1612.683
#> Estimated parameters: 16 
#> AIC = 3257.366
#> BIC = 3320.906; SABIC = 3270.139
#> G2 (239) = 221.19, p = 0.7896
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(gpcmod)
#> $Comfort
#>        a1 ak0 ak1 ak2 ak3 d0    d1    d2    d3
#> par 0.865   0   1   2   3  0 2.831 5.324 3.998
#> 
#> $Work
#>        a1 ak0 ak1 ak2 ak3 d0    d1    d2    d3
#> par 0.841   0   1   2   3  0 1.711 2.578 0.848
#> 
#> $Future
#>        a1 ak0 ak1 ak2 ak3 d0    d1    d2    d3
#> par 2.204   0   1   2   3  0 4.601 6.759 4.918
#> 
#> $Benefit
#>        a1 ak0 ak1 ak2 ak3 d0    d1    d2    d3
#> par 0.724   0   1   2   3  0 2.099 2.899 1.721
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 

# for the nominal model the lowest and highest categories are assumed to be the
#  theoretically lowest and highest categories that related to the latent trait(s)
(nomod <- mirt(Science, 1, 'nominal'))
#> 
Iteration: 1, Log-Lik: -2231.749, Max-Change: 3.43133
Iteration: 2, Log-Lik: -1647.755, Max-Change: 0.78640
Iteration: 3, Log-Lik: -1629.925, Max-Change: 0.48800
Iteration: 4, Log-Lik: -1621.546, Max-Change: 0.44557
Iteration: 5, Log-Lik: -1616.776, Max-Change: 0.30268
Iteration: 6, Log-Lik: -1613.789, Max-Change: 0.29869
Iteration: 7, Log-Lik: -1610.323, Max-Change: 0.20615
Iteration: 8, Log-Lik: -1609.697, Max-Change: 0.28886
Iteration: 9, Log-Lik: -1609.348, Max-Change: 0.13388
Iteration: 10, Log-Lik: -1609.105, Max-Change: 0.11615
Iteration: 11, Log-Lik: -1608.975, Max-Change: 0.10205
Iteration: 12, Log-Lik: -1608.881, Max-Change: 0.09345
Iteration: 13, Log-Lik: -1608.596, Max-Change: 0.06321
Iteration: 14, Log-Lik: -1608.565, Max-Change: 0.03301
Iteration: 15, Log-Lik: -1608.548, Max-Change: 0.02973
Iteration: 16, Log-Lik: -1608.507, Max-Change: 0.01790
Iteration: 17, Log-Lik: -1608.500, Max-Change: 0.02676
Iteration: 18, Log-Lik: -1608.494, Max-Change: 0.02509
Iteration: 19, Log-Lik: -1608.472, Max-Change: 0.00996
Iteration: 20, Log-Lik: -1608.468, Max-Change: 0.00382
Iteration: 21, Log-Lik: -1608.467, Max-Change: 0.00338
Iteration: 22, Log-Lik: -1608.464, Max-Change: 0.03036
Iteration: 23, Log-Lik: -1608.462, Max-Change: 0.00312
Iteration: 24, Log-Lik: -1608.462, Max-Change: 0.00244
Iteration: 25, Log-Lik: -1608.461, Max-Change: 0.00209
Iteration: 26, Log-Lik: -1608.460, Max-Change: 0.00238
Iteration: 27, Log-Lik: -1608.460, Max-Change: 0.00206
Iteration: 28, Log-Lik: -1608.460, Max-Change: 0.02332
Iteration: 29, Log-Lik: -1608.458, Max-Change: 0.00177
Iteration: 30, Log-Lik: -1608.458, Max-Change: 0.00144
Iteration: 31, Log-Lik: -1608.458, Max-Change: 0.00192
Iteration: 32, Log-Lik: -1608.458, Max-Change: 0.00042
Iteration: 33, Log-Lik: -1608.458, Max-Change: 0.00168
Iteration: 34, Log-Lik: -1608.458, Max-Change: 0.00024
Iteration: 35, Log-Lik: -1608.458, Max-Change: 0.00416
Iteration: 36, Log-Lik: -1608.458, Max-Change: 0.00063
Iteration: 37, Log-Lik: -1608.457, Max-Change: 0.00059
Iteration: 38, Log-Lik: -1608.457, Max-Change: 0.00021
Iteration: 39, Log-Lik: -1608.457, Max-Change: 0.00124
Iteration: 40, Log-Lik: -1608.457, Max-Change: 0.00057
Iteration: 41, Log-Lik: -1608.457, Max-Change: 0.00036
Iteration: 42, Log-Lik: -1608.457, Max-Change: 0.00023
Iteration: 43, Log-Lik: -1608.457, Max-Change: 0.02043
Iteration: 44, Log-Lik: -1608.456, Max-Change: 0.00064
Iteration: 45, Log-Lik: -1608.456, Max-Change: 0.00021
Iteration: 46, Log-Lik: -1608.456, Max-Change: 0.00104
Iteration: 47, Log-Lik: -1608.456, Max-Change: 0.00061
Iteration: 48, Log-Lik: -1608.456, Max-Change: 0.00021
Iteration: 49, Log-Lik: -1608.456, Max-Change: 0.00017
Iteration: 50, Log-Lik: -1608.456, Max-Change: 0.00168
Iteration: 51, Log-Lik: -1608.456, Max-Change: 0.00018
Iteration: 52, Log-Lik: -1608.456, Max-Change: 0.00017
Iteration: 53, Log-Lik: -1608.456, Max-Change: 0.00250
Iteration: 54, Log-Lik: -1608.456, Max-Change: 0.00019
Iteration: 55, Log-Lik: -1608.456, Max-Change: 0.00018
Iteration: 56, Log-Lik: -1608.456, Max-Change: 0.02338
Iteration: 57, Log-Lik: -1608.455, Max-Change: 0.00081
Iteration: 58, Log-Lik: -1608.455, Max-Change: 0.00080
Iteration: 59, Log-Lik: -1608.455, Max-Change: 0.00022
Iteration: 60, Log-Lik: -1608.455, Max-Change: 0.00084
Iteration: 61, Log-Lik: -1608.455, Max-Change: 0.00040
Iteration: 62, Log-Lik: -1608.455, Max-Change: 0.00033
Iteration: 63, Log-Lik: -1608.455, Max-Change: 0.00026
Iteration: 64, Log-Lik: -1608.455, Max-Change: 0.00126
Iteration: 65, Log-Lik: -1608.455, Max-Change: 0.00015
Iteration: 66, Log-Lik: -1608.455, Max-Change: 0.00083
Iteration: 67, Log-Lik: -1608.455, Max-Change: 0.00022
Iteration: 68, Log-Lik: -1608.455, Max-Change: 0.00018
Iteration: 69, Log-Lik: -1608.455, Max-Change: 0.00014
Iteration: 70, Log-Lik: -1608.455, Max-Change: 0.00151
Iteration: 71, Log-Lik: -1608.455, Max-Change: 0.00007
#> 
#> Call:
#> mirt(data = Science, model = 1, itemtype = "nominal")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 71 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -1608.455
#> Estimated parameters: 24 
#> AIC = 3264.91
#> BIC = 3360.22; SABIC = 3284.069
#> G2 (231) = 212.73, p = 0.8002
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(nomod) #ordering of ak values suggest that the items are indeed ordinal
#> $Comfort
#>        a1 ak0   ak1   ak2 ak3 d0    d1    d2    d3
#> par 1.008   0 1.541 1.999   3  0 3.639 5.905 4.533
#> 
#> $Work
#>        a1 ak0   ak1 ak2 ak3 d0    d1    d2    d3
#> par 0.841   0 0.689 1.5   3  0 1.464 2.326 0.325
#> 
#> $Future
#>        a1 ak0   ak1   ak2 ak3 d0    d1    d2    d3
#> par 2.041   0 0.762 1.861   3  0 3.668 5.868 3.949
#> 
#> $Benefit
#>        a1 ak0   ak1   ak2 ak3 d0    d1    d2    d3
#> par 0.779   0 1.036 1.742   3  0 2.144 2.911 1.621
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 
anova(gpcmod, nomod)
#>             AIC    SABIC       HQ      BIC    logLik    X2 df    p
#> gpcmod 3257.366 3270.139 3282.549 3320.906 -1612.683              
#> nomod  3264.910 3284.069 3302.684 3360.220 -1608.455 8.456  8 0.39
itemplot(nomod, 3)


# generalized graded unfolding model
(ggum <- mirt(Science, 1, 'ggum'))
#> 
Iteration: 1, Log-Lik: -2388.995, Max-Change: 3.08016
Iteration: 2, Log-Lik: -1679.438, Max-Change: 0.47739
Iteration: 3, Log-Lik: -1652.163, Max-Change: 0.34562
Iteration: 4, Log-Lik: -1642.296, Max-Change: 0.97795
Iteration: 5, Log-Lik: -1636.629, Max-Change: 0.33846
Iteration: 6, Log-Lik: -1633.519, Max-Change: 0.27952
Iteration: 7, Log-Lik: -1628.256, Max-Change: 0.27248
Iteration: 8, Log-Lik: -1627.028, Max-Change: 0.11293
Iteration: 9, Log-Lik: -1626.690, Max-Change: 0.05547
Iteration: 10, Log-Lik: -1626.444, Max-Change: 0.04692
Iteration: 11, Log-Lik: -1626.287, Max-Change: 0.04250
Iteration: 12, Log-Lik: -1626.133, Max-Change: 0.03955
Iteration: 13, Log-Lik: -1625.368, Max-Change: 0.06118
Iteration: 14, Log-Lik: -1625.115, Max-Change: 0.06741
Iteration: 15, Log-Lik: -1624.829, Max-Change: 0.06444
Iteration: 16, Log-Lik: -1623.241, Max-Change: 0.07203
Iteration: 17, Log-Lik: -1622.990, Max-Change: 0.05949
Iteration: 18, Log-Lik: -1622.783, Max-Change: 0.05611
Iteration: 19, Log-Lik: -1622.128, Max-Change: 0.02824
Iteration: 20, Log-Lik: -1622.029, Max-Change: 0.05839
Iteration: 21, Log-Lik: -1621.934, Max-Change: 0.05555
Iteration: 22, Log-Lik: -1621.366, Max-Change: 0.07372
Iteration: 23, Log-Lik: -1621.252, Max-Change: 0.07649
Iteration: 24, Log-Lik: -1621.138, Max-Change: 0.08094
Iteration: 25, Log-Lik: -1620.515, Max-Change: 0.07139
Iteration: 26, Log-Lik: -1620.421, Max-Change: 0.06977
Iteration: 27, Log-Lik: -1620.330, Max-Change: 0.06036
Iteration: 28, Log-Lik: -1619.845, Max-Change: 0.05589
Iteration: 29, Log-Lik: -1619.772, Max-Change: 0.11283
Iteration: 30, Log-Lik: -1619.695, Max-Change: 0.12739
Iteration: 31, Log-Lik: -1619.386, Max-Change: 0.21379
Iteration: 32, Log-Lik: -1619.316, Max-Change: 0.16191
Iteration: 33, Log-Lik: -1619.252, Max-Change: 0.08229
Iteration: 34, Log-Lik: -1619.184, Max-Change: 0.02466
Iteration: 35, Log-Lik: -1619.125, Max-Change: 0.01769
Iteration: 36, Log-Lik: -1619.066, Max-Change: 0.01795
Iteration: 37, Log-Lik: -1618.904, Max-Change: 0.03020
Iteration: 38, Log-Lik: -1618.839, Max-Change: 0.02970
Iteration: 39, Log-Lik: -1618.770, Max-Change: 0.03217
Iteration: 40, Log-Lik: -1618.282, Max-Change: 0.04409
Iteration: 41, Log-Lik: -1618.137, Max-Change: 0.12568
Iteration: 42, Log-Lik: -1617.958, Max-Change: 0.11871
Iteration: 43, Log-Lik: -1616.469, Max-Change: 0.07637
Iteration: 44, Log-Lik: -1616.063, Max-Change: 0.16886
Iteration: 45, Log-Lik: -1615.501, Max-Change: 0.17764
Iteration: 46, Log-Lik: -1613.920, Max-Change: 0.06439
Iteration: 47, Log-Lik: -1613.601, Max-Change: 0.24756
Iteration: 48, Log-Lik: -1613.216, Max-Change: 0.17363
Iteration: 49, Log-Lik: -1612.730, Max-Change: 0.04968
Iteration: 50, Log-Lik: -1612.542, Max-Change: 0.06775
Iteration: 51, Log-Lik: -1612.384, Max-Change: 0.04577
Iteration: 52, Log-Lik: -1612.162, Max-Change: 0.04358
Iteration: 53, Log-Lik: -1612.071, Max-Change: 0.04231
Iteration: 54, Log-Lik: -1611.995, Max-Change: 0.04127
Iteration: 55, Log-Lik: -1611.708, Max-Change: 0.03821
Iteration: 56, Log-Lik: -1611.680, Max-Change: 0.03691
Iteration: 57, Log-Lik: -1611.656, Max-Change: 0.03649
Iteration: 58, Log-Lik: -1611.554, Max-Change: 0.03487
Iteration: 59, Log-Lik: -1611.544, Max-Change: 0.03381
Iteration: 60, Log-Lik: -1611.535, Max-Change: 0.01777
Iteration: 61, Log-Lik: -1611.529, Max-Change: 0.03280
Iteration: 62, Log-Lik: -1611.522, Max-Change: 0.03171
Iteration: 63, Log-Lik: -1611.517, Max-Change: 0.03050
Iteration: 64, Log-Lik: -1611.494, Max-Change: 0.01915
Iteration: 65, Log-Lik: -1611.492, Max-Change: 0.01703
Iteration: 66, Log-Lik: -1611.491, Max-Change: 0.01379
Iteration: 67, Log-Lik: -1611.487, Max-Change: 0.00838
Iteration: 68, Log-Lik: -1611.487, Max-Change: 0.00693
Iteration: 69, Log-Lik: -1611.486, Max-Change: 0.00605
Iteration: 70, Log-Lik: -1611.485, Max-Change: 0.00193
Iteration: 71, Log-Lik: -1611.485, Max-Change: 0.00231
Iteration: 72, Log-Lik: -1611.485, Max-Change: 0.00167
Iteration: 73, Log-Lik: -1611.485, Max-Change: 0.00161
Iteration: 74, Log-Lik: -1611.485, Max-Change: 0.00181
Iteration: 75, Log-Lik: -1611.484, Max-Change: 0.00138
Iteration: 76, Log-Lik: -1611.484, Max-Change: 0.00134
Iteration: 77, Log-Lik: -1611.484, Max-Change: 0.00124
Iteration: 78, Log-Lik: -1611.484, Max-Change: 0.00115
Iteration: 79, Log-Lik: -1611.484, Max-Change: 0.00284
Iteration: 80, Log-Lik: -1611.484, Max-Change: 0.00059
Iteration: 81, Log-Lik: -1611.484, Max-Change: 0.00054
Iteration: 82, Log-Lik: -1611.484, Max-Change: 0.00175
Iteration: 83, Log-Lik: -1611.484, Max-Change: 0.00028
Iteration: 84, Log-Lik: -1611.484, Max-Change: 0.00027
Iteration: 85, Log-Lik: -1611.484, Max-Change: 0.00013
Iteration: 86, Log-Lik: -1611.484, Max-Change: 0.00014
Iteration: 87, Log-Lik: -1611.484, Max-Change: 0.00012
Iteration: 88, Log-Lik: -1611.484, Max-Change: 0.00011
Iteration: 89, Log-Lik: -1611.484, Max-Change: 0.00010
#> 
#> Call:
#> mirt(data = Science, model = 1, itemtype = "ggum")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 89 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: nlminb 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -1611.484
#> Estimated parameters: 20 
#> AIC = 3262.968
#> BIC = 3342.393; SABIC = 3278.934
#> G2 (235) = 218.79, p = 0.7687
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(ggum, simplify=TRUE)
#> $items
#>            a1    b1    t1    t2    t3
#> Comfort 0.824 3.478 6.826 6.475 1.780
#> Work    0.818 3.217 5.280 4.274 0.969
#> Future  2.241 2.800 4.888 3.774 1.961
#> Benefit 0.696 3.584 6.556 4.725 1.744
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
plot(ggum)

plot(ggum, type = 'trace')

plot(ggum, type = 'itemscore')


# monotonic polyomial models
(monopoly <- mirt(Science, 1, 'monopoly'))
#> 
Iteration: 1, Log-Lik: -1632.979, Max-Change: 1.28373
Iteration: 2, Log-Lik: -1611.751, Max-Change: 0.41513
Iteration: 3, Log-Lik: -1606.871, Max-Change: 0.36005
Iteration: 4, Log-Lik: -1604.594, Max-Change: 0.27466
Iteration: 5, Log-Lik: -1603.496, Max-Change: 0.23078
Iteration: 6, Log-Lik: -1602.840, Max-Change: 0.24411
Iteration: 7, Log-Lik: -1602.112, Max-Change: 0.21274
Iteration: 8, Log-Lik: -1601.964, Max-Change: 0.21474
Iteration: 9, Log-Lik: -1601.865, Max-Change: 0.22537
Iteration: 10, Log-Lik: -1601.642, Max-Change: 4.58130
Iteration: 11, Log-Lik: -1601.472, Max-Change: 0.07416
Iteration: 12, Log-Lik: -1601.449, Max-Change: 0.06098
Iteration: 13, Log-Lik: -1601.409, Max-Change: 0.04782
Iteration: 14, Log-Lik: -1601.394, Max-Change: 0.03673
Iteration: 15, Log-Lik: -1601.391, Max-Change: 0.03510
Iteration: 16, Log-Lik: -1601.383, Max-Change: 0.02470
Iteration: 17, Log-Lik: -1601.379, Max-Change: 0.01471
Iteration: 18, Log-Lik: -1601.377, Max-Change: 0.01196
Iteration: 19, Log-Lik: -1601.372, Max-Change: 0.02687
Iteration: 20, Log-Lik: -1601.368, Max-Change: 0.02172
Iteration: 21, Log-Lik: -1601.365, Max-Change: 0.01112
Iteration: 22, Log-Lik: -1601.363, Max-Change: 0.02105
Iteration: 23, Log-Lik: -1601.359, Max-Change: 0.01257
Iteration: 24, Log-Lik: -1601.356, Max-Change: 0.02499
Iteration: 25, Log-Lik: -1601.349, Max-Change: 0.01583
Iteration: 26, Log-Lik: -1601.345, Max-Change: 0.02541
Iteration: 27, Log-Lik: -1601.340, Max-Change: 0.02675
Iteration: 28, Log-Lik: -1601.305, Max-Change: 0.03104
Iteration: 29, Log-Lik: -1601.297, Max-Change: 0.03283
Iteration: 30, Log-Lik: -1601.289, Max-Change: 0.03016
Iteration: 31, Log-Lik: -1601.269, Max-Change: 0.02590
Iteration: 32, Log-Lik: -1601.261, Max-Change: 0.03046
Iteration: 33, Log-Lik: -1601.254, Max-Change: 0.03114
Iteration: 34, Log-Lik: -1601.244, Max-Change: 0.02666
Iteration: 35, Log-Lik: -1601.237, Max-Change: 0.02706
Iteration: 36, Log-Lik: -1601.231, Max-Change: 0.01928
Iteration: 37, Log-Lik: -1601.218, Max-Change: 0.02858
Iteration: 38, Log-Lik: -1601.213, Max-Change: 0.02332
Iteration: 39, Log-Lik: -1601.208, Max-Change: 0.02639
Iteration: 40, Log-Lik: -1601.203, Max-Change: 0.02226
Iteration: 41, Log-Lik: -1601.199, Max-Change: 0.02057
Iteration: 42, Log-Lik: -1601.196, Max-Change: 0.01714
Iteration: 43, Log-Lik: -1601.190, Max-Change: 0.02512
Iteration: 44, Log-Lik: -1601.188, Max-Change: 0.01461
Iteration: 45, Log-Lik: -1601.186, Max-Change: 0.00160
Iteration: 46, Log-Lik: -1601.185, Max-Change: 0.00845
Iteration: 47, Log-Lik: -1601.185, Max-Change: 0.02043
Iteration: 48, Log-Lik: -1601.182, Max-Change: 0.01742
Iteration: 49, Log-Lik: -1601.178, Max-Change: 0.01864
Iteration: 50, Log-Lik: -1601.175, Max-Change: 0.00089
Iteration: 51, Log-Lik: -1601.175, Max-Change: 0.00048
Iteration: 52, Log-Lik: -1601.175, Max-Change: 0.01066
Iteration: 53, Log-Lik: -1601.174, Max-Change: 0.00134
Iteration: 54, Log-Lik: -1601.174, Max-Change: 0.00062
Iteration: 55, Log-Lik: -1601.174, Max-Change: 0.00007
#> 
#> Call:
#> mirt(data = Science, model = 1, itemtype = "monopoly")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 55 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -1601.174
#> Estimated parameters: 24 
#> AIC = 3250.347
#> BIC = 3345.657; SABIC = 3269.506
#> G2 (231) = 198.17, p = 0.9424
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(monopoly, simplify=TRUE)
#> $items
#>          omega   xi1   xi2    xi3 alpha1   tau2
#> Comfort -1.431 2.911 2.218 -1.469 -0.934  0.728
#> Work    -0.412 1.378 0.698 -2.152 -0.499 -1.151
#> Future   0.833 4.988 2.259 -1.910  0.019 -8.472
#> Benefit -1.714 1.883 0.618 -1.389 -1.424  0.716
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
plot(monopoly)

plot(monopoly, type = 'trace')

plot(monopoly, type = 'itemscore')


# unipolar IRT model
unimod <- mirt(Science, itemtype = 'ULL')
#> 
Iteration: 1, Log-Lik: -1644.943, Max-Change: 0.41474
Iteration: 2, Log-Lik: -1618.453, Max-Change: 0.32498
Iteration: 3, Log-Lik: -1611.142, Max-Change: 0.23517
Iteration: 4, Log-Lik: -1608.577, Max-Change: 0.15407
Iteration: 5, Log-Lik: -1607.588, Max-Change: 0.12836
Iteration: 6, Log-Lik: -1607.050, Max-Change: 0.15376
Iteration: 7, Log-Lik: -1606.229, Max-Change: 0.07291
Iteration: 8, Log-Lik: -1606.031, Max-Change: 0.05626
Iteration: 9, Log-Lik: -1605.916, Max-Change: 0.05430
Iteration: 10, Log-Lik: -1605.695, Max-Change: 0.03203
Iteration: 11, Log-Lik: -1605.672, Max-Change: 0.02436
Iteration: 12, Log-Lik: -1605.661, Max-Change: 0.02554
Iteration: 13, Log-Lik: -1605.637, Max-Change: 0.01607
Iteration: 14, Log-Lik: -1605.632, Max-Change: 0.01219
Iteration: 15, Log-Lik: -1605.628, Max-Change: 0.01067
Iteration: 16, Log-Lik: -1605.622, Max-Change: 0.00325
Iteration: 17, Log-Lik: -1605.622, Max-Change: 0.00018
Iteration: 18, Log-Lik: -1605.622, Max-Change: 0.01898
Iteration: 19, Log-Lik: -1605.617, Max-Change: 0.00551
Iteration: 20, Log-Lik: -1605.616, Max-Change: 0.00039
Iteration: 21, Log-Lik: -1605.616, Max-Change: 0.00395
Iteration: 22, Log-Lik: -1605.616, Max-Change: 0.00118
Iteration: 23, Log-Lik: -1605.616, Max-Change: 0.00176
Iteration: 24, Log-Lik: -1605.616, Max-Change: 0.00056
Iteration: 25, Log-Lik: -1605.616, Max-Change: 0.00284
Iteration: 26, Log-Lik: -1605.616, Max-Change: 0.00019
Iteration: 27, Log-Lik: -1605.616, Max-Change: 0.00026
Iteration: 28, Log-Lik: -1605.616, Max-Change: 0.00019
Iteration: 29, Log-Lik: -1605.616, Max-Change: 0.00039
Iteration: 30, Log-Lik: -1605.616, Max-Change: 0.00014
Iteration: 31, Log-Lik: -1605.616, Max-Change: 0.00061
Iteration: 32, Log-Lik: -1605.616, Max-Change: 0.00024
Iteration: 33, Log-Lik: -1605.616, Max-Change: 0.00051
Iteration: 34, Log-Lik: -1605.616, Max-Change: 0.00105
Iteration: 35, Log-Lik: -1605.616, Max-Change: 0.00023
Iteration: 36, Log-Lik: -1605.616, Max-Change: 0.00050
Iteration: 37, Log-Lik: -1605.616, Max-Change: 0.00042
Iteration: 38, Log-Lik: -1605.616, Max-Change: 0.00021
Iteration: 39, Log-Lik: -1605.616, Max-Change: 0.00058
Iteration: 40, Log-Lik: -1605.616, Max-Change: 0.00021
Iteration: 41, Log-Lik: -1605.616, Max-Change: 0.00047
Iteration: 42, Log-Lik: -1605.616, Max-Change: 0.00018
Iteration: 43, Log-Lik: -1605.616, Max-Change: 0.00080
Iteration: 44, Log-Lik: -1605.616, Max-Change: 0.00018
Iteration: 45, Log-Lik: -1605.616, Max-Change: 0.00035
Iteration: 46, Log-Lik: -1605.616, Max-Change: 0.00030
Iteration: 47, Log-Lik: -1605.616, Max-Change: 0.00074
Iteration: 48, Log-Lik: -1605.616, Max-Change: 0.00017
Iteration: 49, Log-Lik: -1605.616, Max-Change: 0.00015
Iteration: 50, Log-Lik: -1605.616, Max-Change: 0.00029
Iteration: 51, Log-Lik: -1605.616, Max-Change: 0.00067
Iteration: 52, Log-Lik: -1605.616, Max-Change: 0.00038
Iteration: 53, Log-Lik: -1605.616, Max-Change: 0.00023
Iteration: 54, Log-Lik: -1605.616, Max-Change: 0.00042
Iteration: 55, Log-Lik: -1605.616, Max-Change: 0.00037
Iteration: 56, Log-Lik: -1605.616, Max-Change: 0.00051
Iteration: 57, Log-Lik: -1605.616, Max-Change: 0.00029
Iteration: 58, Log-Lik: -1605.615, Max-Change: 0.00022
Iteration: 59, Log-Lik: -1605.615, Max-Change: 0.00041
Iteration: 60, Log-Lik: -1605.615, Max-Change: 0.00014
Iteration: 61, Log-Lik: -1605.615, Max-Change: 0.00011
Iteration: 62, Log-Lik: -1605.615, Max-Change: 0.00073
Iteration: 63, Log-Lik: -1605.615, Max-Change: 0.00093
Iteration: 64, Log-Lik: -1605.615, Max-Change: 0.00044
Iteration: 65, Log-Lik: -1605.615, Max-Change: 0.00017
Iteration: 66, Log-Lik: -1605.615, Max-Change: 0.00038
Iteration: 67, Log-Lik: -1605.615, Max-Change: 0.00020
Iteration: 68, Log-Lik: -1605.615, Max-Change: 0.00036
Iteration: 69, Log-Lik: -1605.615, Max-Change: 0.00016
Iteration: 70, Log-Lik: -1605.615, Max-Change: 0.00069
Iteration: 71, Log-Lik: -1605.615, Max-Change: 0.00023
Iteration: 72, Log-Lik: -1605.615, Max-Change: 0.00035
Iteration: 73, Log-Lik: -1605.615, Max-Change: 0.00048
Iteration: 74, Log-Lik: -1605.615, Max-Change: 0.00018
Iteration: 75, Log-Lik: -1605.615, Max-Change: 0.00041
Iteration: 76, Log-Lik: -1605.615, Max-Change: 0.00023
Iteration: 77, Log-Lik: -1605.615, Max-Change: 0.00043
Iteration: 78, Log-Lik: -1605.615, Max-Change: 0.00018
Iteration: 79, Log-Lik: -1605.615, Max-Change: 0.00015
Iteration: 80, Log-Lik: -1605.615, Max-Change: 0.00035
Iteration: 81, Log-Lik: -1605.615, Max-Change: 0.00041
Iteration: 82, Log-Lik: -1605.615, Max-Change: 0.00040
Iteration: 83, Log-Lik: -1605.615, Max-Change: 0.00015
Iteration: 84, Log-Lik: -1605.615, Max-Change: 0.00018
Iteration: 85, Log-Lik: -1605.615, Max-Change: 0.00040
Iteration: 86, Log-Lik: -1605.615, Max-Change: 0.00064
Iteration: 87, Log-Lik: -1605.615, Max-Change: 0.00027
Iteration: 88, Log-Lik: -1605.615, Max-Change: 0.00020
Iteration: 89, Log-Lik: -1605.615, Max-Change: 0.00026
Iteration: 90, Log-Lik: -1605.615, Max-Change: 0.00016
Iteration: 91, Log-Lik: -1605.615, Max-Change: 0.00012
Iteration: 92, Log-Lik: -1605.615, Max-Change: 0.00111
Iteration: 93, Log-Lik: -1605.615, Max-Change: 0.00052
Iteration: 94, Log-Lik: -1605.615, Max-Change: 0.00040
Iteration: 95, Log-Lik: -1605.615, Max-Change: 0.00011
Iteration: 96, Log-Lik: -1605.615, Max-Change: 0.00016
Iteration: 97, Log-Lik: -1605.615, Max-Change: 0.00024
Iteration: 98, Log-Lik: -1605.615, Max-Change: 0.00044
Iteration: 99, Log-Lik: -1605.615, Max-Change: 0.00015
Iteration: 100, Log-Lik: -1605.615, Max-Change: 0.00012
Iteration: 101, Log-Lik: -1605.615, Max-Change: 0.00018
Iteration: 102, Log-Lik: -1605.615, Max-Change: 0.00050
Iteration: 103, Log-Lik: -1605.615, Max-Change: 0.00012
Iteration: 104, Log-Lik: -1605.615, Max-Change: 0.00014
Iteration: 105, Log-Lik: -1605.615, Max-Change: 0.00050
Iteration: 106, Log-Lik: -1605.615, Max-Change: 0.00015
Iteration: 107, Log-Lik: -1605.615, Max-Change: 0.00019
Iteration: 108, Log-Lik: -1605.615, Max-Change: 0.00012
Iteration: 109, Log-Lik: -1605.615, Max-Change: 0.00044
Iteration: 110, Log-Lik: -1605.615, Max-Change: 0.00020
Iteration: 111, Log-Lik: -1605.615, Max-Change: 0.00027
Iteration: 112, Log-Lik: -1605.615, Max-Change: 0.00115
Iteration: 113, Log-Lik: -1605.615, Max-Change: 0.00030
Iteration: 114, Log-Lik: -1605.615, Max-Change: 0.00014
Iteration: 115, Log-Lik: -1605.615, Max-Change: 0.00041
Iteration: 116, Log-Lik: -1605.615, Max-Change: 0.00052
Iteration: 117, Log-Lik: -1605.615, Max-Change: 0.00020
Iteration: 118, Log-Lik: -1605.615, Max-Change: 0.00017
Iteration: 119, Log-Lik: -1605.615, Max-Change: 0.00038
Iteration: 120, Log-Lik: -1605.615, Max-Change: 0.00011
Iteration: 121, Log-Lik: -1605.615, Max-Change: 0.00047
Iteration: 122, Log-Lik: -1605.615, Max-Change: 0.00020
Iteration: 123, Log-Lik: -1605.615, Max-Change: 0.00041
Iteration: 124, Log-Lik: -1605.615, Max-Change: 0.00033
Iteration: 125, Log-Lik: -1605.615, Max-Change: 0.00013
Iteration: 126, Log-Lik: -1605.615, Max-Change: 0.00024
Iteration: 127, Log-Lik: -1605.615, Max-Change: 0.00024
Iteration: 128, Log-Lik: -1605.615, Max-Change: 0.00052
Iteration: 129, Log-Lik: -1605.615, Max-Change: 0.00017
Iteration: 130, Log-Lik: -1605.615, Max-Change: 0.00014
Iteration: 131, Log-Lik: -1605.615, Max-Change: 0.00027
Iteration: 132, Log-Lik: -1605.615, Max-Change: 0.00012
Iteration: 133, Log-Lik: -1605.615, Max-Change: 0.00049
Iteration: 134, Log-Lik: -1605.615, Max-Change: 0.00017
Iteration: 135, Log-Lik: -1605.615, Max-Change: 0.00031
Iteration: 136, Log-Lik: -1605.615, Max-Change: 0.00034
Iteration: 137, Log-Lik: -1605.615, Max-Change: 0.00014
Iteration: 138, Log-Lik: -1605.615, Max-Change: 0.00031
Iteration: 139, Log-Lik: -1605.615, Max-Change: 0.00023
Iteration: 140, Log-Lik: -1605.615, Max-Change: 0.00009
coef(unimod, simplify=TRUE)
#> $items
#>          eta1 log_lambda1 log_lambda2 log_lambda3
#> Comfort 1.175       4.776       2.299      -1.709
#> Work    1.618       2.533       0.554      -2.736
#> Future  2.801       4.030       1.525      -2.594
#> Benefit 1.319       3.020       0.681      -1.995
#> 
#> $GroupPars
#>     meanlog sdlog
#> par       0     1
#> 
plot(unimod)

plot(unimod, type = 'trace')

itemplot(unimod, 1)


# following use the correct log-normal density for latent trait
itemfit(unimod)
#>      item   S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
#> 1 Comfort  5.664       6      0.000  0.462
#> 2    Work 10.136       8      0.026  0.256
#> 3  Future 19.477       8      0.061  0.013
#> 4 Benefit 12.106      11      0.016  0.356
M2(unimod, type = 'C2')
#>             M2 df            p     RMSEA    RMSEA_5  RMSEA_95     SRMSR
#> stats 18.69535  2 8.716799e-05 0.1461148 0.09026025 0.2096111 0.0786373
#>             TLI       CFI
#> stats 0.7376985 0.9125662
fs <- fscores(unimod)
hist(fs, 20)

fscores(unimod, method = 'EAPsum', full.scores = FALSE)
#>       df     X2 p.X2 SEM.alpha rxx.alpha rxx_F1
#> stats 10 33.926    0     1.305     0.658  0.531
#> 
#>    Sum.Scores    F1 SE_F1 observed expected std.res
#> 4           4 0.138 0.153        2    0.127   5.251
#> 5           5 0.304 0.088        1    0.766   0.267
#> 6           6 0.328 0.084        2    4.339   1.123
#> 7           7 0.352 0.126        1   13.909   3.461
#> 8           8 0.407 0.199       11   27.739   3.178
#> 9           9 0.530 0.305       32   40.624   1.353
#> 10         10 0.748 0.440       58   52.271   0.792
#> 11         11 1.053 0.604       70   63.507   0.815
#> 12         12 1.478 0.845       91   68.879   2.665
#> 13         13 2.164 1.282       56   54.418   0.214
#> 14         14 3.299 2.001       36   36.187   0.031
#> 15         15 5.109 3.236       20   20.821   0.180
#> 16         16 8.222 5.298       12    8.414   1.236

## example applying survey weights.
# weight the first half of the cases to be more representative of population
survey.weights <- c(rep(2, nrow(Science)/2), rep(1, nrow(Science)/2))
survey.weights <- survey.weights/sum(survey.weights) * nrow(Science)
unweighted <- mirt(Science, 1)
#> 
Iteration: 1, Log-Lik: -1629.361, Max-Change: 0.50660
Iteration: 2, Log-Lik: -1617.374, Max-Change: 0.25442
Iteration: 3, Log-Lik: -1612.894, Max-Change: 0.16991
Iteration: 4, Log-Lik: -1610.306, Max-Change: 0.10461
Iteration: 5, Log-Lik: -1609.814, Max-Change: 0.09162
Iteration: 6, Log-Lik: -1609.534, Max-Change: 0.07363
Iteration: 7, Log-Lik: -1609.030, Max-Change: 0.03677
Iteration: 8, Log-Lik: -1608.988, Max-Change: 0.03200
Iteration: 9, Log-Lik: -1608.958, Max-Change: 0.02754
Iteration: 10, Log-Lik: -1608.878, Max-Change: 0.01443
Iteration: 11, Log-Lik: -1608.875, Max-Change: 0.00847
Iteration: 12, Log-Lik: -1608.873, Max-Change: 0.00515
Iteration: 13, Log-Lik: -1608.872, Max-Change: 0.00550
Iteration: 14, Log-Lik: -1608.872, Max-Change: 0.00318
Iteration: 15, Log-Lik: -1608.871, Max-Change: 0.00462
Iteration: 16, Log-Lik: -1608.871, Max-Change: 0.00277
Iteration: 17, Log-Lik: -1608.870, Max-Change: 0.00145
Iteration: 18, Log-Lik: -1608.870, Max-Change: 0.00175
Iteration: 19, Log-Lik: -1608.870, Max-Change: 0.00126
Iteration: 20, Log-Lik: -1608.870, Max-Change: 0.00025
Iteration: 21, Log-Lik: -1608.870, Max-Change: 0.00285
Iteration: 22, Log-Lik: -1608.870, Max-Change: 0.00108
Iteration: 23, Log-Lik: -1608.870, Max-Change: 0.00022
Iteration: 24, Log-Lik: -1608.870, Max-Change: 0.00059
Iteration: 25, Log-Lik: -1608.870, Max-Change: 0.00014
Iteration: 26, Log-Lik: -1608.870, Max-Change: 0.00068
Iteration: 27, Log-Lik: -1608.870, Max-Change: 0.00065
Iteration: 28, Log-Lik: -1608.870, Max-Change: 0.00019
Iteration: 29, Log-Lik: -1608.870, Max-Change: 0.00061
Iteration: 30, Log-Lik: -1608.870, Max-Change: 0.00012
Iteration: 31, Log-Lik: -1608.870, Max-Change: 0.00012
Iteration: 32, Log-Lik: -1608.870, Max-Change: 0.00058
Iteration: 33, Log-Lik: -1608.870, Max-Change: 0.00055
Iteration: 34, Log-Lik: -1608.870, Max-Change: 0.00015
Iteration: 35, Log-Lik: -1608.870, Max-Change: 0.00052
Iteration: 36, Log-Lik: -1608.870, Max-Change: 0.00010
weighted <- mirt(Science, 1, survey.weights=survey.weights)
#> 
Iteration: 1, Log-Lik: -1645.766, Max-Change: 0.53685
Iteration: 2, Log-Lik: -1636.301, Max-Change: 0.20495
Iteration: 3, Log-Lik: -1632.918, Max-Change: 0.16001
Iteration: 4, Log-Lik: -1630.851, Max-Change: 0.10241
Iteration: 5, Log-Lik: -1630.399, Max-Change: 0.07562
Iteration: 6, Log-Lik: -1630.135, Max-Change: 0.06727
Iteration: 7, Log-Lik: -1629.625, Max-Change: 0.04208
Iteration: 8, Log-Lik: -1629.557, Max-Change: 0.03806
Iteration: 9, Log-Lik: -1629.503, Max-Change: 0.03490
Iteration: 10, Log-Lik: -1629.325, Max-Change: 0.02305
Iteration: 11, Log-Lik: -1629.316, Max-Change: 0.01321
Iteration: 12, Log-Lik: -1629.310, Max-Change: 0.01669
Iteration: 13, Log-Lik: -1629.292, Max-Change: 0.00712
Iteration: 14, Log-Lik: -1629.289, Max-Change: 0.01075
Iteration: 15, Log-Lik: -1629.288, Max-Change: 0.00655
Iteration: 16, Log-Lik: -1629.285, Max-Change: 0.00332
Iteration: 17, Log-Lik: -1629.285, Max-Change: 0.00763
Iteration: 18, Log-Lik: -1629.284, Max-Change: 0.00312
Iteration: 19, Log-Lik: -1629.284, Max-Change: 0.00421
Iteration: 20, Log-Lik: -1629.284, Max-Change: 0.00658
Iteration: 21, Log-Lik: -1629.283, Max-Change: 0.00218
Iteration: 22, Log-Lik: -1629.283, Max-Change: 0.00284
Iteration: 23, Log-Lik: -1629.283, Max-Change: 0.00568
Iteration: 24, Log-Lik: -1629.282, Max-Change: 0.00177
Iteration: 25, Log-Lik: -1629.282, Max-Change: 0.00161
Iteration: 26, Log-Lik: -1629.282, Max-Change: 0.00184
Iteration: 27, Log-Lik: -1629.282, Max-Change: 0.00039
Iteration: 28, Log-Lik: -1629.282, Max-Change: 0.00037
Iteration: 29, Log-Lik: -1629.282, Max-Change: 0.00134
Iteration: 30, Log-Lik: -1629.282, Max-Change: 0.00512
Iteration: 31, Log-Lik: -1629.282, Max-Change: 0.00112
Iteration: 32, Log-Lik: -1629.282, Max-Change: 0.00029
Iteration: 33, Log-Lik: -1629.282, Max-Change: 0.00080
Iteration: 34, Log-Lik: -1629.282, Max-Change: 0.00017
Iteration: 35, Log-Lik: -1629.282, Max-Change: 0.00148
Iteration: 36, Log-Lik: -1629.282, Max-Change: 0.00083
Iteration: 37, Log-Lik: -1629.282, Max-Change: 0.00018
Iteration: 38, Log-Lik: -1629.282, Max-Change: 0.00074
Iteration: 39, Log-Lik: -1629.282, Max-Change: 0.00076
Iteration: 40, Log-Lik: -1629.282, Max-Change: 0.00108
Iteration: 41, Log-Lik: -1629.282, Max-Change: 0.00019
Iteration: 42, Log-Lik: -1629.282, Max-Change: 0.00061
Iteration: 43, Log-Lik: -1629.281, Max-Change: 0.00014
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Iteration: 47, Log-Lik: -1629.281, Max-Change: 0.00057
Iteration: 48, Log-Lik: -1629.281, Max-Change: 0.00060
Iteration: 49, Log-Lik: -1629.281, Max-Change: 0.00016
Iteration: 50, Log-Lik: -1629.281, Max-Change: 0.00057
Iteration: 51, Log-Lik: -1629.281, Max-Change: 0.00058
Iteration: 52, Log-Lik: -1629.281, Max-Change: 0.00016
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Iteration: 54, Log-Lik: -1629.281, Max-Change: 0.00011
Iteration: 55, Log-Lik: -1629.281, Max-Change: 0.00054
Iteration: 56, Log-Lik: -1629.281, Max-Change: 0.00018
Iteration: 57, Log-Lik: -1629.281, Max-Change: 0.00011
Iteration: 58, Log-Lik: -1629.281, Max-Change: 0.00046
Iteration: 59, Log-Lik: -1629.281, Max-Change: 0.00049
Iteration: 60, Log-Lik: -1629.281, Max-Change: 0.00013
Iteration: 61, Log-Lik: -1629.281, Max-Change: 0.00012
Iteration: 62, Log-Lik: -1629.281, Max-Change: 0.00043
Iteration: 63, Log-Lik: -1629.281, Max-Change: 0.00046
Iteration: 64, Log-Lik: -1629.281, Max-Change: 0.00014
Iteration: 65, Log-Lik: -1629.281, Max-Change: 0.00044
Iteration: 66, Log-Lik: -1629.281, Max-Change: 0.00009

###########
# empirical dimensionality testing that includes 'guessing'

data(SAT12)
data <- key2binary(SAT12,
  key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
itemstats(data)
#> $overall
#>    N mean_total.score sd_total.score ave.r  sd.r alpha SEM.alpha
#>  600           18.202          5.054 0.108 0.075 0.798     2.272
#> 
#> $itemstats
#>           N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item.1  600 0.283 0.451   0.380         0.300       0.793
#> Item.2  600 0.568 0.496   0.539         0.464       0.785
#> Item.3  600 0.280 0.449   0.446         0.371       0.789
#> Item.4  600 0.378 0.485   0.325         0.235       0.796
#> Item.5  600 0.620 0.486   0.424         0.340       0.791
#> Item.6  600 0.160 0.367   0.414         0.351       0.791
#> Item.7  600 0.760 0.427   0.366         0.289       0.793
#> Item.8  600 0.202 0.402   0.307         0.233       0.795
#> Item.9  600 0.885 0.319   0.189         0.127       0.798
#> Item.10 600 0.422 0.494   0.465         0.383       0.789
#> Item.11 600 0.983 0.128   0.181         0.156       0.797
#> Item.12 600 0.415 0.493   0.173         0.076       0.803
#> Item.13 600 0.662 0.474   0.438         0.358       0.790
#> Item.14 600 0.723 0.448   0.411         0.333       0.791
#> Item.15 600 0.817 0.387   0.393         0.325       0.792
#> Item.16 600 0.413 0.493   0.367         0.278       0.794
#> Item.17 600 0.963 0.188   0.238         0.202       0.796
#> Item.18 600 0.352 0.478   0.576         0.508       0.783
#> Item.19 600 0.548 0.498   0.401         0.314       0.792
#> Item.20 600 0.873 0.333   0.376         0.318       0.792
#> Item.21 600 0.915 0.279   0.190         0.136       0.798
#> Item.22 600 0.935 0.247   0.284         0.238       0.795
#> Item.23 600 0.313 0.464   0.338         0.253       0.795
#> Item.24 600 0.728 0.445   0.422         0.346       0.791
#> Item.25 600 0.375 0.485   0.383         0.297       0.793
#> Item.26 600 0.460 0.499   0.562         0.489       0.783
#> Item.27 600 0.862 0.346   0.425         0.367       0.791
#> Item.28 600 0.530 0.500   0.465         0.383       0.789
#> Item.29 600 0.340 0.474   0.407         0.324       0.791
#> Item.30 600 0.440 0.497   0.255         0.159       0.799
#> Item.31 600 0.833 0.373   0.479         0.419       0.788
#> Item.32 600 0.162 0.368   0.110         0.037       0.802
#> 
#> $proportions
#>             0     1
#> Item.1  0.717 0.283
#> Item.2  0.432 0.568
#> Item.3  0.720 0.280
#> Item.4  0.622 0.378
#> Item.5  0.380 0.620
#> Item.6  0.840 0.160
#> Item.7  0.240 0.760
#> Item.8  0.798 0.202
#> Item.9  0.115 0.885
#> Item.10 0.578 0.422
#> Item.11 0.017 0.983
#> Item.12 0.585 0.415
#> Item.13 0.338 0.662
#> Item.14 0.277 0.723
#> Item.15 0.183 0.817
#> Item.16 0.587 0.413
#> Item.17 0.037 0.963
#> Item.18 0.648 0.352
#> Item.19 0.452 0.548
#> Item.20 0.127 0.873
#> Item.21 0.085 0.915
#> Item.22 0.065 0.935
#> Item.23 0.687 0.313
#> Item.24 0.272 0.728
#> Item.25 0.625 0.375
#> Item.26 0.540 0.460
#> Item.27 0.138 0.862
#> Item.28 0.470 0.530
#> Item.29 0.660 0.340
#> Item.30 0.560 0.440
#> Item.31 0.167 0.833
#> Item.32 0.838 0.162
#> 

mod1 <- mirt(data, 1)
#> 
Iteration: 1, Log-Lik: -9647.510, Max-Change: 0.81958
Iteration: 2, Log-Lik: -9501.953, Max-Change: 0.61213
Iteration: 3, Log-Lik: -9491.324, Max-Change: 0.30029
Iteration: 4, Log-Lik: -9489.773, Max-Change: 0.17556
Iteration: 5, Log-Lik: -9489.311, Max-Change: 0.05283
Iteration: 6, Log-Lik: -9489.122, Max-Change: 0.03138
Iteration: 7, Log-Lik: -9489.045, Max-Change: 0.01853
Iteration: 8, Log-Lik: -9489.000, Max-Change: 0.01225
Iteration: 9, Log-Lik: -9488.978, Max-Change: 0.00708
Iteration: 10, Log-Lik: -9488.964, Max-Change: 0.00300
Iteration: 11, Log-Lik: -9488.961, Max-Change: 0.00284
Iteration: 12, Log-Lik: -9488.959, Max-Change: 0.00186
Iteration: 13, Log-Lik: -9488.958, Max-Change: 0.00149
Iteration: 14, Log-Lik: -9488.957, Max-Change: 0.00118
Iteration: 15, Log-Lik: -9488.956, Max-Change: 0.00173
Iteration: 16, Log-Lik: -9488.956, Max-Change: 0.00086
Iteration: 17, Log-Lik: -9488.955, Max-Change: 0.00026
Iteration: 18, Log-Lik: -9488.955, Max-Change: 0.00020
Iteration: 19, Log-Lik: -9488.955, Max-Change: 0.00017
Iteration: 20, Log-Lik: -9488.955, Max-Change: 0.00015
Iteration: 21, Log-Lik: -9488.955, Max-Change: 0.00014
Iteration: 22, Log-Lik: -9488.955, Max-Change: 0.00045
Iteration: 23, Log-Lik: -9488.955, Max-Change: 0.00004
extract.mirt(mod1, 'time') #time elapsed for each estimation component
#> TOTAL:   Data  Estep  Mstep     SE   Post 
#>  0.238  0.033  0.076  0.118  0.000  0.001 

# optionally use Newton-Raphson for (generally) faster convergence in the M-step's
mod1 <- mirt(data, 1, optimizer = 'NR')
#> 
Iteration: 1, Log-Lik: -9647.510, Max-Change: 0.79951
Iteration: 2, Log-Lik: -9503.605, Max-Change: 0.35290
Iteration: 3, Log-Lik: -9491.858, Max-Change: 0.17654
Iteration: 4, Log-Lik: -9490.044, Max-Change: 0.09576
Iteration: 5, Log-Lik: -9489.448, Max-Change: 0.05539
Iteration: 6, Log-Lik: -9489.192, Max-Change: 0.03370
Iteration: 7, Log-Lik: -9489.012, Max-Change: 0.01254
Iteration: 8, Log-Lik: -9488.984, Max-Change: 0.00869
Iteration: 9, Log-Lik: -9488.970, Max-Change: 0.00607
Iteration: 10, Log-Lik: -9488.957, Max-Change: 0.00186
Iteration: 11, Log-Lik: -9488.956, Max-Change: 0.00134
Iteration: 12, Log-Lik: -9488.956, Max-Change: 0.00096
Iteration: 13, Log-Lik: -9488.955, Max-Change: 0.00026
Iteration: 14, Log-Lik: -9488.955, Max-Change: 0.00019
Iteration: 15, Log-Lik: -9488.955, Max-Change: 0.00016
Iteration: 16, Log-Lik: -9488.955, Max-Change: 0.00009
extract.mirt(mod1, 'time')
#> TOTAL:   Data  Estep  Mstep     SE   Post 
#>  0.238  0.035  0.089  0.071  0.000  0.001 

mod2 <- mirt(data, 2, optimizer = 'NR')
#> 
Iteration: 1, Log-Lik: -10018.384, Max-Change: 0.61117
Iteration: 2, Log-Lik: -9561.695, Max-Change: 0.40673
Iteration: 3, Log-Lik: -9487.442, Max-Change: 0.26447
Iteration: 4, Log-Lik: -9463.109, Max-Change: 0.17883
Iteration: 5, Log-Lik: -9452.978, Max-Change: 0.12468
Iteration: 6, Log-Lik: -9448.191, Max-Change: 0.08906
Iteration: 7, Log-Lik: -9445.721, Max-Change: 0.06487
Iteration: 8, Log-Lik: -9444.354, Max-Change: 0.04801
Iteration: 9, Log-Lik: -9443.550, Max-Change: 0.03602
Iteration: 10, Log-Lik: -9442.377, Max-Change: 0.01473
Iteration: 11, Log-Lik: -9442.274, Max-Change: 0.01272
Iteration: 12, Log-Lik: -9442.199, Max-Change: 0.01100
Iteration: 13, Log-Lik: -9441.989, Max-Change: 0.00244
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#> Warning: EM cycles terminated after 500 iterations.
# difficulty converging with reduced quadpts, reduce TOL
mod3 <- mirt(data, 3, TOL = .001, optimizer = 'NR')
#> 
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Iteration: 325, Log-Lik: -9415.315, Max-Change: 0.00150
Iteration: 326, Log-Lik: -9415.313, Max-Change: 0.00150
Iteration: 327, Log-Lik: -9415.312, Max-Change: 0.00150
Iteration: 328, Log-Lik: -9415.305, Max-Change: 0.00149
Iteration: 329, Log-Lik: -9415.303, Max-Change: 0.00149
Iteration: 330, Log-Lik: -9415.302, Max-Change: 0.00149
Iteration: 331, Log-Lik: -9415.295, Max-Change: 0.00148
Iteration: 332, Log-Lik: -9415.294, Max-Change: 0.00148
Iteration: 333, Log-Lik: -9415.292, Max-Change: 0.00148
Iteration: 334, Log-Lik: -9415.285, Max-Change: 0.00148
Iteration: 335, Log-Lik: -9415.284, Max-Change: 0.00147
Iteration: 336, Log-Lik: -9415.283, Max-Change: 0.00147
Iteration: 337, Log-Lik: -9415.275, Max-Change: 0.00147
Iteration: 338, Log-Lik: -9415.274, Max-Change: 0.00147
Iteration: 339, Log-Lik: -9415.273, Max-Change: 0.00147
Iteration: 340, Log-Lik: -9415.266, Max-Change: 0.00146
Iteration: 341, Log-Lik: -9415.264, Max-Change: 0.00146
Iteration: 342, Log-Lik: -9415.263, Max-Change: 0.00146
Iteration: 343, Log-Lik: -9415.256, Max-Change: 0.00145
Iteration: 344, Log-Lik: -9415.255, Max-Change: 0.00145
Iteration: 345, Log-Lik: -9415.253, Max-Change: 0.00144
Iteration: 346, Log-Lik: -9415.246, Max-Change: 0.00143
Iteration: 347, Log-Lik: -9415.245, Max-Change: 0.00143
Iteration: 348, Log-Lik: -9415.244, Max-Change: 0.00143
Iteration: 349, Log-Lik: -9415.237, Max-Change: 0.00142
Iteration: 350, Log-Lik: -9415.236, Max-Change: 0.00142
Iteration: 351, Log-Lik: -9415.234, Max-Change: 0.00142
Iteration: 352, Log-Lik: -9415.227, Max-Change: 0.00141
Iteration: 353, Log-Lik: -9415.226, Max-Change: 0.00141
Iteration: 354, Log-Lik: -9415.225, Max-Change: 0.00141
Iteration: 355, Log-Lik: -9415.218, Max-Change: 0.00139
Iteration: 356, Log-Lik: -9415.217, Max-Change: 0.00139
Iteration: 357, Log-Lik: -9415.216, Max-Change: 0.00139
Iteration: 358, Log-Lik: -9415.209, Max-Change: 0.00138
Iteration: 359, Log-Lik: -9415.208, Max-Change: 0.00138
Iteration: 360, Log-Lik: -9415.207, Max-Change: 0.00137
Iteration: 361, Log-Lik: -9415.200, Max-Change: 0.00136
Iteration: 362, Log-Lik: -9415.199, Max-Change: 0.00136
Iteration: 363, Log-Lik: -9415.198, Max-Change: 0.00136
Iteration: 364, Log-Lik: -9415.192, Max-Change: 0.00134
Iteration: 365, Log-Lik: -9415.191, Max-Change: 0.00134
Iteration: 366, Log-Lik: -9415.189, Max-Change: 0.00134
Iteration: 367, Log-Lik: -9415.183, Max-Change: 0.00132
Iteration: 368, Log-Lik: -9415.182, Max-Change: 0.00132
Iteration: 369, Log-Lik: -9415.181, Max-Change: 0.00132
Iteration: 370, Log-Lik: -9415.175, Max-Change: 0.00130
Iteration: 371, Log-Lik: -9415.174, Max-Change: 0.00130
Iteration: 372, Log-Lik: -9415.173, Max-Change: 0.00130
Iteration: 373, Log-Lik: -9415.167, Max-Change: 0.00128
Iteration: 374, Log-Lik: -9415.166, Max-Change: 0.00128
Iteration: 375, Log-Lik: -9415.165, Max-Change: 0.00128
Iteration: 376, Log-Lik: -9415.159, Max-Change: 0.00126
Iteration: 377, Log-Lik: -9415.158, Max-Change: 0.00126
Iteration: 378, Log-Lik: -9415.157, Max-Change: 0.00126
Iteration: 379, Log-Lik: -9415.151, Max-Change: 0.00124
Iteration: 380, Log-Lik: -9415.150, Max-Change: 0.00124
Iteration: 381, Log-Lik: -9415.150, Max-Change: 0.00123
Iteration: 382, Log-Lik: -9415.144, Max-Change: 0.00121
Iteration: 383, Log-Lik: -9415.143, Max-Change: 0.00121
Iteration: 384, Log-Lik: -9415.142, Max-Change: 0.00121
Iteration: 385, Log-Lik: -9415.137, Max-Change: 0.00119
Iteration: 386, Log-Lik: -9415.136, Max-Change: 0.00119
Iteration: 387, Log-Lik: -9415.135, Max-Change: 0.00119
Iteration: 388, Log-Lik: -9415.130, Max-Change: 0.00117
Iteration: 389, Log-Lik: -9415.130, Max-Change: 0.00116
Iteration: 390, Log-Lik: -9415.129, Max-Change: 0.00116
Iteration: 391, Log-Lik: -9415.124, Max-Change: 0.00114
Iteration: 392, Log-Lik: -9415.123, Max-Change: 0.00114
Iteration: 393, Log-Lik: -9415.122, Max-Change: 0.00113
Iteration: 394, Log-Lik: -9415.118, Max-Change: 0.00112
Iteration: 395, Log-Lik: -9415.117, Max-Change: 0.00111
Iteration: 396, Log-Lik: -9415.116, Max-Change: 0.00111
Iteration: 397, Log-Lik: -9415.112, Max-Change: 0.00109
Iteration: 398, Log-Lik: -9415.111, Max-Change: 0.00109
Iteration: 399, Log-Lik: -9415.110, Max-Change: 0.00108
Iteration: 400, Log-Lik: -9415.106, Max-Change: 0.00106
Iteration: 401, Log-Lik: -9415.106, Max-Change: 0.00106
Iteration: 402, Log-Lik: -9415.105, Max-Change: 0.00106
Iteration: 403, Log-Lik: -9415.101, Max-Change: 0.00104
Iteration: 404, Log-Lik: -9415.100, Max-Change: 0.00103
Iteration: 405, Log-Lik: -9415.100, Max-Change: 0.00103
Iteration: 406, Log-Lik: -9415.096, Max-Change: 0.00101
Iteration: 407, Log-Lik: -9415.095, Max-Change: 0.00101
Iteration: 408, Log-Lik: -9415.095, Max-Change: 0.00100
Iteration: 409, Log-Lik: -9415.091, Max-Change: 0.00098
anova(mod1,mod2)
#>           AIC    SABIC       HQ      BIC    logLik     X2 df p
#> mod1 19105.91 19184.13 19215.46 19387.31 -9488.955            
#> mod2 19073.92 19190.03 19236.53 19491.63 -9441.963 93.985 31 0
anova(mod2, mod3) #negative AIC, 2 factors probably best
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> mod2 19073.92 19190.03 19236.53 19491.63 -9441.963                
#> mod3 19080.18 19232.96 19294.13 19629.80 -9415.090 53.744 30 0.005

# same as above, but using the QMCEM method for generally better accuracy in mod3
mod3 <- mirt(data, 3, method = 'QMCEM', TOL = .001, optimizer = 'NR')
#> 
Iteration: 1, Log-Lik: -10713.240, Max-Change: 2.06007
Iteration: 2, Log-Lik: -9590.755, Max-Change: 0.39362
Iteration: 3, Log-Lik: -9480.999, Max-Change: 0.26003
Iteration: 4, Log-Lik: -9448.491, Max-Change: 0.17920
Iteration: 5, Log-Lik: -9435.600, Max-Change: 0.12685
Iteration: 6, Log-Lik: -9429.583, Max-Change: 0.09154
Iteration: 7, Log-Lik: -9426.428, Max-Change: 0.06709
Iteration: 8, Log-Lik: -9424.610, Max-Change: 0.04983
Iteration: 9, Log-Lik: -9423.474, Max-Change: 0.03747
Iteration: 10, Log-Lik: -9421.426, Max-Change: 0.02045
Iteration: 11, Log-Lik: -9421.210, Max-Change: 0.01976
Iteration: 12, Log-Lik: -9421.032, Max-Change: 0.01901
Iteration: 13, Log-Lik: -9420.346, Max-Change: 0.01348
Iteration: 14, Log-Lik: -9420.299, Max-Change: 0.01321
Iteration: 15, Log-Lik: -9420.256, Max-Change: 0.01289
Iteration: 16, Log-Lik: -9420.045, Max-Change: 0.01081
Iteration: 17, Log-Lik: -9420.019, Max-Change: 0.01056
Iteration: 18, Log-Lik: -9419.994, Max-Change: 0.01035
Iteration: 19, Log-Lik: -9419.851, Max-Change: 0.00913
Iteration: 20, Log-Lik: -9419.829, Max-Change: 0.00910
Iteration: 21, Log-Lik: -9419.807, Max-Change: 0.00906
Iteration: 22, Log-Lik: -9419.676, Max-Change: 0.00861
Iteration: 23, Log-Lik: -9419.654, Max-Change: 0.00854
Iteration: 24, Log-Lik: -9419.633, Max-Change: 0.00847
Iteration: 25, Log-Lik: -9419.502, Max-Change: 0.00790
Iteration: 26, Log-Lik: -9419.480, Max-Change: 0.00787
Iteration: 27, Log-Lik: -9419.458, Max-Change: 0.00783
Iteration: 28, Log-Lik: -9419.325, Max-Change: 0.00728
Iteration: 29, Log-Lik: -9419.302, Max-Change: 0.00723
Iteration: 30, Log-Lik: -9419.279, Max-Change: 0.00716
Iteration: 31, Log-Lik: -9419.138, Max-Change: 0.00643
Iteration: 32, Log-Lik: -9419.113, Max-Change: 0.00638
Iteration: 33, Log-Lik: -9419.088, Max-Change: 0.00631
Iteration: 34, Log-Lik: -9418.933, Max-Change: 0.00621
Iteration: 35, Log-Lik: -9418.906, Max-Change: 0.00622
Iteration: 36, Log-Lik: -9418.877, Max-Change: 0.00624
Iteration: 37, Log-Lik: -9418.699, Max-Change: 0.00645
Iteration: 38, Log-Lik: -9418.666, Max-Change: 0.00648
Iteration: 39, Log-Lik: -9418.633, Max-Change: 0.00652
Iteration: 40, Log-Lik: -9418.422, Max-Change: 0.00694
Iteration: 41, Log-Lik: -9418.382, Max-Change: 0.00698
Iteration: 42, Log-Lik: -9418.342, Max-Change: 0.00705
Iteration: 43, Log-Lik: -9418.090, Max-Change: 0.00760
Iteration: 44, Log-Lik: -9418.043, Max-Change: 0.00763
Iteration: 45, Log-Lik: -9417.997, Max-Change: 0.00768
Iteration: 46, Log-Lik: -9417.708, Max-Change: 0.00805
Iteration: 47, Log-Lik: -9417.658, Max-Change: 0.00804
Iteration: 48, Log-Lik: -9417.608, Max-Change: 0.00803
Iteration: 49, Log-Lik: -9417.313, Max-Change: 0.00800
Iteration: 50, Log-Lik: -9417.266, Max-Change: 0.00791
Iteration: 51, Log-Lik: -9417.221, Max-Change: 0.00783
Iteration: 52, Log-Lik: -9416.963, Max-Change: 0.00800
Iteration: 53, Log-Lik: -9416.926, Max-Change: 0.00821
Iteration: 54, Log-Lik: -9416.891, Max-Change: 0.00820
Iteration: 55, Log-Lik: -9416.700, Max-Change: 0.00746
Iteration: 56, Log-Lik: -9416.675, Max-Change: 0.00781
Iteration: 57, Log-Lik: -9416.651, Max-Change: 0.00774
Iteration: 58, Log-Lik: -9416.521, Max-Change: 0.00632
Iteration: 59, Log-Lik: -9416.504, Max-Change: 0.00661
Iteration: 60, Log-Lik: -9416.487, Max-Change: 0.00653
Iteration: 61, Log-Lik: -9416.395, Max-Change: 0.00528
Iteration: 62, Log-Lik: -9416.382, Max-Change: 0.00536
Iteration: 63, Log-Lik: -9416.370, Max-Change: 0.00529
Iteration: 64, Log-Lik: -9416.299, Max-Change: 0.00437
Iteration: 65, Log-Lik: -9416.288, Max-Change: 0.00433
Iteration: 66, Log-Lik: -9416.278, Max-Change: 0.00428
Iteration: 67, Log-Lik: -9416.219, Max-Change: 0.00367
Iteration: 68, Log-Lik: -9416.210, Max-Change: 0.00366
Iteration: 69, Log-Lik: -9416.201, Max-Change: 0.00365
Iteration: 70, Log-Lik: -9416.151, Max-Change: 0.00353
Iteration: 71, Log-Lik: -9416.143, Max-Change: 0.00352
Iteration: 72, Log-Lik: -9416.135, Max-Change: 0.00351
Iteration: 73, Log-Lik: -9416.092, Max-Change: 0.00338
Iteration: 74, Log-Lik: -9416.086, Max-Change: 0.00337
Iteration: 75, Log-Lik: -9416.079, Max-Change: 0.00335
Iteration: 76, Log-Lik: -9416.042, Max-Change: 0.00322
Iteration: 77, Log-Lik: -9416.037, Max-Change: 0.00320
Iteration: 78, Log-Lik: -9416.031, Max-Change: 0.00318
Iteration: 79, Log-Lik: -9416.000, Max-Change: 0.00305
Iteration: 80, Log-Lik: -9415.995, Max-Change: 0.00303
Iteration: 81, Log-Lik: -9415.990, Max-Change: 0.00301
Iteration: 82, Log-Lik: -9415.964, Max-Change: 0.00287
Iteration: 83, Log-Lik: -9415.960, Max-Change: 0.00285
Iteration: 84, Log-Lik: -9415.956, Max-Change: 0.00283
Iteration: 85, Log-Lik: -9415.934, Max-Change: 0.00270
Iteration: 86, Log-Lik: -9415.931, Max-Change: 0.00268
Iteration: 87, Log-Lik: -9415.928, Max-Change: 0.00266
Iteration: 88, Log-Lik: -9415.909, Max-Change: 0.00254
Iteration: 89, Log-Lik: -9415.906, Max-Change: 0.00252
Iteration: 90, Log-Lik: -9415.904, Max-Change: 0.00250
Iteration: 91, Log-Lik: -9415.888, Max-Change: 0.00238
Iteration: 92, Log-Lik: -9415.886, Max-Change: 0.00236
Iteration: 93, Log-Lik: -9415.884, Max-Change: 0.00234
Iteration: 94, Log-Lik: -9415.871, Max-Change: 0.00223
Iteration: 95, Log-Lik: -9415.869, Max-Change: 0.00221
Iteration: 96, Log-Lik: -9415.867, Max-Change: 0.00219
Iteration: 97, Log-Lik: -9415.856, Max-Change: 0.00208
Iteration: 98, Log-Lik: -9415.854, Max-Change: 0.00207
Iteration: 99, Log-Lik: -9415.853, Max-Change: 0.00205
Iteration: 100, Log-Lik: -9415.844, Max-Change: 0.00195
Iteration: 101, Log-Lik: -9415.843, Max-Change: 0.00193
Iteration: 102, Log-Lik: -9415.841, Max-Change: 0.00191
Iteration: 103, Log-Lik: -9415.834, Max-Change: 0.00182
Iteration: 104, Log-Lik: -9415.833, Max-Change: 0.00180
Iteration: 105, Log-Lik: -9415.832, Max-Change: 0.00179
Iteration: 106, Log-Lik: -9415.825, Max-Change: 0.00170
Iteration: 107, Log-Lik: -9415.824, Max-Change: 0.00168
Iteration: 108, Log-Lik: -9415.823, Max-Change: 0.00167
Iteration: 109, Log-Lik: -9415.818, Max-Change: 0.00158
Iteration: 110, Log-Lik: -9415.817, Max-Change: 0.00157
Iteration: 111, Log-Lik: -9415.817, Max-Change: 0.00156
Iteration: 112, Log-Lik: -9415.812, Max-Change: 0.00147
Iteration: 113, Log-Lik: -9415.811, Max-Change: 0.00146
Iteration: 114, Log-Lik: -9415.811, Max-Change: 0.00145
Iteration: 115, Log-Lik: -9415.807, Max-Change: 0.00137
Iteration: 116, Log-Lik: -9415.807, Max-Change: 0.00136
Iteration: 117, Log-Lik: -9415.806, Max-Change: 0.00135
Iteration: 118, Log-Lik: -9415.803, Max-Change: 0.00128
Iteration: 119, Log-Lik: -9415.802, Max-Change: 0.00127
Iteration: 120, Log-Lik: -9415.802, Max-Change: 0.00125
Iteration: 121, Log-Lik: -9415.799, Max-Change: 0.00119
Iteration: 122, Log-Lik: -9415.799, Max-Change: 0.00118
Iteration: 123, Log-Lik: -9415.799, Max-Change: 0.00117
Iteration: 124, Log-Lik: -9415.796, Max-Change: 0.00110
Iteration: 125, Log-Lik: -9415.796, Max-Change: 0.00109
Iteration: 126, Log-Lik: -9415.796, Max-Change: 0.00108
Iteration: 127, Log-Lik: -9415.794, Max-Change: 0.00103
Iteration: 128, Log-Lik: -9415.794, Max-Change: 0.00102
Iteration: 129, Log-Lik: -9415.793, Max-Change: 0.00101
Iteration: 130, Log-Lik: -9415.792, Max-Change: 0.00095
anova(mod2, mod3)
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> mod2 19073.92 19190.03 19236.53 19491.63 -9441.963                
#> mod3 19081.58 19234.36 19295.54 19631.20 -9415.792 52.342 30 0.007

# with fixed guessing parameters
mod1g <- mirt(data, 1, guess = .1)
#> 
Iteration: 1, Log-Lik: -9675.656, Max-Change: 1.64000
Iteration: 2, Log-Lik: -9492.451, Max-Change: 1.00116
Iteration: 3, Log-Lik: -9473.154, Max-Change: 0.21779
Iteration: 4, Log-Lik: -9467.592, Max-Change: 0.14237
Iteration: 5, Log-Lik: -9464.833, Max-Change: 0.08381
Iteration: 6, Log-Lik: -9463.303, Max-Change: 0.05844
Iteration: 7, Log-Lik: -9461.487, Max-Change: 0.02865
Iteration: 8, Log-Lik: -9461.247, Max-Change: 0.02380
Iteration: 9, Log-Lik: -9461.087, Max-Change: 0.01951
Iteration: 10, Log-Lik: -9460.790, Max-Change: 0.01025
Iteration: 11, Log-Lik: -9460.760, Max-Change: 0.00739
Iteration: 12, Log-Lik: -9460.742, Max-Change: 0.00457
Iteration: 13, Log-Lik: -9460.724, Max-Change: 0.00451
Iteration: 14, Log-Lik: -9460.716, Max-Change: 0.00308
Iteration: 15, Log-Lik: -9460.711, Max-Change: 0.00235
Iteration: 16, Log-Lik: -9460.702, Max-Change: 0.00250
Iteration: 17, Log-Lik: -9460.700, Max-Change: 0.00163
Iteration: 18, Log-Lik: -9460.699, Max-Change: 0.00162
Iteration: 19, Log-Lik: -9460.697, Max-Change: 0.00100
Iteration: 20, Log-Lik: -9460.697, Max-Change: 0.00031
Iteration: 21, Log-Lik: -9460.696, Max-Change: 0.00023
Iteration: 22, Log-Lik: -9460.696, Max-Change: 0.00022
Iteration: 23, Log-Lik: -9460.696, Max-Change: 0.00020
Iteration: 24, Log-Lik: -9460.696, Max-Change: 0.00019
Iteration: 25, Log-Lik: -9460.696, Max-Change: 0.00071
Iteration: 26, Log-Lik: -9460.696, Max-Change: 0.00055
Iteration: 27, Log-Lik: -9460.696, Max-Change: 0.00054
Iteration: 28, Log-Lik: -9460.696, Max-Change: 0.00019
Iteration: 29, Log-Lik: -9460.696, Max-Change: 0.00039
Iteration: 30, Log-Lik: -9460.696, Max-Change: 0.00012
Iteration: 31, Log-Lik: -9460.696, Max-Change: 0.00009
coef(mod1g)
#> $Item.1
#>        a1      d   g u
#> par 1.211 -1.737 0.1 1
#> 
#> $Item.2
#>       a1     d   g u
#> par 1.78 0.147 0.1 1
#> 
#> $Item.3
#>       a1    d   g u
#> par 1.91 -2.2 0.1 1
#> 
#> $Item.4
#>        a1      d   g u
#> par 0.833 -0.944 0.1 1
#> 
#> $Item.5
#>        a1     d   g u
#> par 1.089 0.399 0.1 1
#> 
#> $Item.6
#>        a1      d   g u
#> par 3.265 -5.212 0.1 1
#> 
#> $Item.7
#>       a1     d   g u
#> par 1.02 1.224 0.1 1
#> 
#> $Item.8
#>        a1      d   g u
#> par 1.639 -2.977 0.1 1
#> 
#> $Item.9
#>       a1     d   g u
#> par 0.49 2.007 0.1 1
#> 
#> $Item.10
#>        a1      d   g u
#> par 1.257 -0.756 0.1 1
#> 
#> $Item.11
#>       a1    d   g u
#> par 1.68 5.18 0.1 1
#> 
#> $Item.12
#>        a1      d   g u
#> par 0.191 -0.625 0.1 1
#> 
#> $Item.13
#>        a1     d   g u
#> par 1.147 0.654 0.1 1
#> 
#> $Item.14
#>        a1     d   g u
#> par 1.099 1.008 0.1 1
#> 
#> $Item.15
#>        a1    d   g u
#> par 1.337 1.79 0.1 1
#> 
#> $Item.16
#>        a1      d   g u
#> par 0.923 -0.744 0.1 1
#> 
#> $Item.17
#>        a1     d   g u
#> par 1.519 4.077 0.1 1
#> 
#> $Item.18
#>        a1      d   g u
#> par 2.585 -1.749 0.1 1
#> 
#> $Item.19
#>       a1      d   g u
#> par 0.91 -0.002 0.1 1
#> 
#> $Item.20
#>        a1     d   g u
#> par 1.485 2.438 0.1 1
#> 
#> $Item.21
#>        a1     d   g u
#> par 0.616 2.407 0.1 1
#> 
#> $Item.22
#>        a1     d   g u
#> par 1.429 3.291 0.1 1
#> 
#> $Item.23
#>       a1      d   g u
#> par 0.96 -1.393 0.1 1
#> 
#> $Item.24
#>        a1     d   g u
#> par 1.282 1.099 0.1 1
#> 
#> $Item.25
#>        a1  d   g u
#> par 1.028 -1 0.1 1
#> 
#> $Item.26
#>        a1      d   g u
#> par 2.059 -0.658 0.1 1
#> 
#> $Item.27
#>        a1     d   g u
#> par 1.839 2.564 0.1 1
#> 
#> $Item.28
#>        a1      d   g u
#> par 1.222 -0.095 0.1 1
#> 
#> $Item.29
#>        a1      d   g u
#> par 1.281 -1.357 0.1 1
#> 
#> $Item.30
#>        a1      d   g u
#> par 0.444 -0.521 0.1 1
#> 
#> $Item.31
#>        a1     d   g u
#> par 2.476 2.697 0.1 1
#> 
#> $Item.32
#>        a1      d   g u
#> par 0.461 -2.742 0.1 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 

###########
# graded rating scale example

# make some data
set.seed(1234)
a <- matrix(rep(1, 10))
d <- matrix(c(1,0.5,-.5,-1), 10, 4, byrow = TRUE)
c <- seq(-1, 1, length.out=10)
data <- simdata(a, d + c, 2000, itemtype = rep('graded',10))
itemstats(data)
#> $overall
#>     N mean_total.score sd_total.score ave.r  sd.r alpha SEM.alpha
#>  2000           20.196           8.33 0.203 0.027 0.719     4.419
#> 
#> $itemstats
#>            N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item_1  2000 1.284 1.510   0.512         0.359       0.700
#> Item_2  2000 1.427 1.544   0.529         0.375       0.697
#> Item_3  2000 1.592 1.584   0.545         0.389       0.695
#> Item_4  2000 1.774 1.586   0.538         0.381       0.696
#> Item_5  2000 1.910 1.607   0.539         0.380       0.696
#> Item_6  2000 2.124 1.606   0.533         0.373       0.697
#> Item_7  2000 2.284 1.598   0.520         0.359       0.700
#> Item_8  2000 2.420 1.583   0.578         0.430       0.688
#> Item_9  2000 2.606 1.543   0.530         0.377       0.697
#> Item_10 2000 2.776 1.491   0.495         0.342       0.702
#> 
#> $proportions
#>             0     1     2     3     4
#> Item_1  0.500 0.096 0.182 0.065 0.158
#> Item_2  0.450 0.108 0.197 0.059 0.187
#> Item_3  0.407 0.108 0.182 0.092 0.212
#> Item_4  0.346 0.111 0.212 0.085 0.246
#> Item_5  0.319 0.102 0.211 0.086 0.281
#> Item_6  0.269 0.097 0.205 0.099 0.330
#> Item_7  0.244 0.073 0.211 0.101 0.372
#> Item_8  0.216 0.074 0.195 0.106 0.410
#> Item_9  0.175 0.072 0.196 0.083 0.473
#> Item_10 0.150 0.059 0.174 0.102 0.516
#> 

mod1 <- mirt(data, 1)
#> 
Iteration: 1, Log-Lik: -27605.030, Max-Change: 0.15290
Iteration: 2, Log-Lik: -27583.637, Max-Change: 0.08050
Iteration: 3, Log-Lik: -27578.624, Max-Change: 0.05002
Iteration: 4, Log-Lik: -27576.935, Max-Change: 0.02227
Iteration: 5, Log-Lik: -27576.312, Max-Change: 0.01248
Iteration: 6, Log-Lik: -27576.123, Max-Change: 0.00712
Iteration: 7, Log-Lik: -27576.046, Max-Change: 0.00272
Iteration: 8, Log-Lik: -27576.035, Max-Change: 0.00156
Iteration: 9, Log-Lik: -27576.030, Max-Change: 0.00078
Iteration: 10, Log-Lik: -27576.028, Max-Change: 0.00044
Iteration: 11, Log-Lik: -27576.028, Max-Change: 0.00016
Iteration: 12, Log-Lik: -27576.028, Max-Change: 0.00029
Iteration: 13, Log-Lik: -27576.028, Max-Change: 0.00031
Iteration: 14, Log-Lik: -27576.027, Max-Change: 0.00006
mod2 <- mirt(data, 1, itemtype = 'grsm')
#> 
Iteration: 1, Log-Lik: -29266.482, Max-Change: 2.06018
Iteration: 2, Log-Lik: -27606.775, Max-Change: 0.12306
Iteration: 3, Log-Lik: -27597.848, Max-Change: 0.03037
Iteration: 4, Log-Lik: -27596.914, Max-Change: 0.01114
Iteration: 5, Log-Lik: -27596.870, Max-Change: 0.00076
Iteration: 6, Log-Lik: -27596.866, Max-Change: 0.00052
Iteration: 7, Log-Lik: -27596.863, Max-Change: 0.00016
Iteration: 8, Log-Lik: -27596.863, Max-Change: 0.00036
Iteration: 9, Log-Lik: -27596.863, Max-Change: 0.00011
Iteration: 10, Log-Lik: -27596.863, Max-Change: 0.00008
coef(mod2)
#> $Item_1
#>        a1    b1     b2     b3     b4 c
#> par 0.959 0.001 -0.507 -1.541 -2.032 0
#> 
#> $Item_2
#>        a1    b1     b2     b3     b4     c
#> par 0.987 0.001 -0.507 -1.541 -2.032 0.235
#> 
#> $Item_3
#>        a1    b1     b2     b3     b4     c
#> par 0.994 0.001 -0.507 -1.541 -2.032 0.457
#> 
#> $Item_4
#>        a1    b1     b2     b3     b4     c
#> par 1.027 0.001 -0.507 -1.541 -2.032 0.728
#> 
#> $Item_5
#>        a1    b1     b2     b3     b4     c
#> par 0.995 0.001 -0.507 -1.541 -2.032 0.895
#> 
#> $Item_6
#>        a1    b1     b2     b3     b4     c
#> par 0.987 0.001 -0.507 -1.541 -2.032 1.179
#> 
#> $Item_7
#>        a1    b1     b2     b3     b4     c
#> par 0.957 0.001 -0.507 -1.541 -2.032 1.404
#> 
#> $Item_8
#>       a1    b1     b2     b3     b4     c
#> par 1.04 0.001 -0.507 -1.541 -2.032 1.578
#> 
#> $Item_9
#>        a1    b1     b2     b3     b4     c
#> par 0.964 0.001 -0.507 -1.541 -2.032 1.878
#> 
#> $Item_10
#>        a1    b1     b2     b3     b4     c
#> par 0.947 0.001 -0.507 -1.541 -2.032 2.136
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0      1
#> 
anova(mod2, mod1) #not sig, mod2 should be preferred
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> mod2 55239.72 55295.47 55287.03 55368.55 -27596.86                
#> mod1 55252.05 55373.25 55354.88 55532.10 -27576.03 41.671 27 0.035
itemplot(mod2, 1)

itemplot(mod2, 5)

itemplot(mod2, 10)


###########
# 2PL nominal response model example (Suh and Bolt, 2010)
data(SAT12)
SAT12[SAT12 == 8] <- NA #set 8 as a missing value
head(SAT12)
#>   Item.1 Item.2 Item.3 Item.4 Item.5 Item.6 Item.7 Item.8 Item.9 Item.10
#> 1      1      4      5      2      3      1      2      1      3       1
#> 2      3      4      2     NA      3      3      2     NA      3       1
#> 3      1      4      5      4      3      2      2      3      3       2
#> 4      2      4      4      2      3      3      2      4      3       2
#> 5      2      4      5      2      3      2      2      1      1       2
#> 6      1      4      3      1      3      2      2      3      3       1
#>   Item.11 Item.12 Item.13 Item.14 Item.15 Item.16 Item.17 Item.18 Item.19
#> 1       2       4       2       1       5       3       4       4       1
#> 2       2      NA       2       1       5       2       4       1       1
#> 3       2       1       3       1       5       5       4       1       3
#> 4       2       4       2       1       5       2       4       1       3
#> 5       2       4       2       1       5       4       4       5       1
#> 6       2       3       2       1       5       5       4       4       1
#>   Item.20 Item.21 Item.22 Item.23 Item.24 Item.25 Item.26 Item.27 Item.28
#> 1       4       3       3       4       1       3       5       1       3
#> 2       4       3       3      NA       1      NA       4       1       4
#> 3       4       3       3       1       1       3       4       1       3
#> 4       4       3       1       5       2       5       4       1       3
#> 5       4       3       3       3       1       1       5       1       3
#> 6       4       3       3       4       1       1       4       1       4
#>   Item.29 Item.30 Item.31 Item.32
#> 1       1       5       4       5
#> 2       5      NA       4      NA
#> 3       4       4       4       1
#> 4       4       2       4       2
#> 5       1       2       4       1
#> 6       2       3       4       3

# correct answer key
key <- c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5)
scoredSAT12 <- key2binary(SAT12, key)
mod0 <- mirt(scoredSAT12, 1)
#> 
Iteration: 1, Log-Lik: -9613.786, Max-Change: 0.80344
Iteration: 2, Log-Lik: -9468.739, Max-Change: 0.52487
Iteration: 3, Log-Lik: -9458.099, Max-Change: 0.23219
Iteration: 4, Log-Lik: -9456.607, Max-Change: 0.11785
Iteration: 5, Log-Lik: -9456.201, Max-Change: 0.04589
Iteration: 6, Log-Lik: -9456.025, Max-Change: 0.02606
Iteration: 7, Log-Lik: -9455.927, Max-Change: 0.01620
Iteration: 8, Log-Lik: -9455.891, Max-Change: 0.00925
Iteration: 9, Log-Lik: -9455.874, Max-Change: 0.00465
Iteration: 10, Log-Lik: -9455.861, Max-Change: 0.00390
Iteration: 11, Log-Lik: -9455.857, Max-Change: 0.00267
Iteration: 12, Log-Lik: -9455.854, Max-Change: 0.00191
Iteration: 13, Log-Lik: -9455.851, Max-Change: 0.00214
Iteration: 14, Log-Lik: -9455.850, Max-Change: 0.00163
Iteration: 15, Log-Lik: -9455.850, Max-Change: 0.00071
Iteration: 16, Log-Lik: -9455.849, Max-Change: 0.00036
Iteration: 17, Log-Lik: -9455.849, Max-Change: 0.00033
Iteration: 18, Log-Lik: -9455.849, Max-Change: 0.00029
Iteration: 19, Log-Lik: -9455.849, Max-Change: 0.00075
Iteration: 20, Log-Lik: -9455.849, Max-Change: 0.00007

# for first 5 items use 2PLNRM and nominal
scoredSAT12[,1:5] <- as.matrix(SAT12[,1:5])
mod1 <- mirt(scoredSAT12, 1, c(rep('nominal',5),rep('2PL', 27)))
#> 
Iteration: 1, Log-Lik: -13260.607, Max-Change: 3.74948
Iteration: 2, Log-Lik: -11857.044, Max-Change: 3.70653
Iteration: 3, Log-Lik: -11688.923, Max-Change: 2.83639
Iteration: 4, Log-Lik: -11640.311, Max-Change: 1.97611
Iteration: 5, Log-Lik: -11621.311, Max-Change: 1.05067
Iteration: 6, Log-Lik: -11614.142, Max-Change: 0.85011
Iteration: 7, Log-Lik: -11610.708, Max-Change: 0.40399
Iteration: 8, Log-Lik: -11608.938, Max-Change: 0.98591
Iteration: 9, Log-Lik: -11607.717, Max-Change: 0.30236
Iteration: 10, Log-Lik: -11607.162, Max-Change: 5.01139
Iteration: 11, Log-Lik: -11604.918, Max-Change: 0.23936
Iteration: 12, Log-Lik: -11604.331, Max-Change: 1.24402
Iteration: 13, Log-Lik: -11603.909, Max-Change: 0.84659
Iteration: 14, Log-Lik: -11603.554, Max-Change: 0.56149
Iteration: 15, Log-Lik: -11603.392, Max-Change: 0.00074
Iteration: 16, Log-Lik: -11603.392, Max-Change: 0.00074
Iteration: 17, Log-Lik: -11603.390, Max-Change: 6.41921
Iteration: 18, Log-Lik: -11602.678, Max-Change: 0.15371
Iteration: 19, Log-Lik: -11602.668, Max-Change: 1.09968
Iteration: 20, Log-Lik: -11602.392, Max-Change: 0.05449
Iteration: 21, Log-Lik: -11602.339, Max-Change: 6.76908
Iteration: 22, Log-Lik: -11602.031, Max-Change: 0.11257
Iteration: 23, Log-Lik: -11601.914, Max-Change: 0.00027
Iteration: 24, Log-Lik: -11601.911, Max-Change: 0.00008
mod2 <- mirt(scoredSAT12, 1, c(rep('2PLNRM',5),rep('2PL', 27)), key=key)
#> 
Iteration: 1, Log-Lik: -12010.931, Max-Change: 3.39278
Iteration: 2, Log-Lik: -11636.213, Max-Change: 1.60456
Iteration: 3, Log-Lik: -11604.353, Max-Change: 0.23662
Iteration: 4, Log-Lik: -11600.473, Max-Change: 0.11573
Iteration: 5, Log-Lik: -11599.708, Max-Change: 0.07663
Iteration: 6, Log-Lik: -11599.489, Max-Change: 0.03183
Iteration: 7, Log-Lik: -11599.413, Max-Change: 0.02481
Iteration: 8, Log-Lik: -11599.377, Max-Change: 0.01140
Iteration: 9, Log-Lik: -11599.358, Max-Change: 0.00739
Iteration: 10, Log-Lik: -11599.346, Max-Change: 0.00406
Iteration: 11, Log-Lik: -11599.340, Max-Change: 0.00221
Iteration: 12, Log-Lik: -11599.339, Max-Change: 0.00181
Iteration: 13, Log-Lik: -11599.336, Max-Change: 0.00057
Iteration: 14, Log-Lik: -11599.336, Max-Change: 0.00043
Iteration: 15, Log-Lik: -11599.336, Max-Change: 0.00042
Iteration: 16, Log-Lik: -11599.336, Max-Change: 0.00039
Iteration: 17, Log-Lik: -11599.336, Max-Change: 0.00190
Iteration: 18, Log-Lik: -11599.335, Max-Change: 0.00034
Iteration: 19, Log-Lik: -11599.335, Max-Change: 0.00030
Iteration: 20, Log-Lik: -11599.335, Max-Change: 0.00142
Iteration: 21, Log-Lik: -11599.335, Max-Change: 0.00054
Iteration: 22, Log-Lik: -11599.335, Max-Change: 0.00027
Iteration: 23, Log-Lik: -11599.335, Max-Change: 0.00104
Iteration: 24, Log-Lik: -11599.335, Max-Change: 0.00063
Iteration: 25, Log-Lik: -11599.335, Max-Change: 0.00024
Iteration: 26, Log-Lik: -11599.335, Max-Change: 0.00081
Iteration: 27, Log-Lik: -11599.335, Max-Change: 0.00057
Iteration: 28, Log-Lik: -11599.335, Max-Change: 0.00020
Iteration: 29, Log-Lik: -11599.335, Max-Change: 0.00066
Iteration: 30, Log-Lik: -11599.335, Max-Change: 0.00047
Iteration: 31, Log-Lik: -11599.335, Max-Change: 0.00017
Iteration: 32, Log-Lik: -11599.335, Max-Change: 0.00069
Iteration: 33, Log-Lik: -11599.335, Max-Change: 0.00040
Iteration: 34, Log-Lik: -11599.335, Max-Change: 0.00014
Iteration: 35, Log-Lik: -11599.335, Max-Change: 0.00070
Iteration: 36, Log-Lik: -11599.335, Max-Change: 0.00035
Iteration: 37, Log-Lik: -11599.335, Max-Change: 0.00014
Iteration: 38, Log-Lik: -11599.335, Max-Change: 0.00069
Iteration: 39, Log-Lik: -11599.335, Max-Change: 0.00031
Iteration: 40, Log-Lik: -11599.335, Max-Change: 0.00014
Iteration: 41, Log-Lik: -11599.335, Max-Change: 0.00069
Iteration: 42, Log-Lik: -11599.335, Max-Change: 0.00028
Iteration: 43, Log-Lik: -11599.335, Max-Change: 0.00014
Iteration: 44, Log-Lik: -11599.335, Max-Change: 0.00067
Iteration: 45, Log-Lik: -11599.335, Max-Change: 0.00027
Iteration: 46, Log-Lik: -11599.335, Max-Change: 0.00013
Iteration: 47, Log-Lik: -11599.335, Max-Change: 0.00066
Iteration: 48, Log-Lik: -11599.335, Max-Change: 0.00025
Iteration: 49, Log-Lik: -11599.335, Max-Change: 0.00013
Iteration: 50, Log-Lik: -11599.335, Max-Change: 0.00065
Iteration: 51, Log-Lik: -11599.335, Max-Change: 0.00024
Iteration: 52, Log-Lik: -11599.335, Max-Change: 0.00013
Iteration: 53, Log-Lik: -11599.335, Max-Change: 0.00063
Iteration: 54, Log-Lik: -11599.335, Max-Change: 0.00023
Iteration: 55, Log-Lik: -11599.335, Max-Change: 0.00012
Iteration: 56, Log-Lik: -11599.335, Max-Change: 0.00062
Iteration: 57, Log-Lik: -11599.335, Max-Change: 0.00022
Iteration: 58, Log-Lik: -11599.335, Max-Change: 0.00012
Iteration: 59, Log-Lik: -11599.335, Max-Change: 0.00060
Iteration: 60, Log-Lik: -11599.335, Max-Change: 0.00021
Iteration: 61, Log-Lik: -11599.335, Max-Change: 0.00012
Iteration: 62, Log-Lik: -11599.335, Max-Change: 0.00059
Iteration: 63, Log-Lik: -11599.335, Max-Change: 0.00021
Iteration: 64, Log-Lik: -11599.335, Max-Change: 0.00012
Iteration: 65, Log-Lik: -11599.335, Max-Change: 0.00058
Iteration: 66, Log-Lik: -11599.335, Max-Change: 0.00020
Iteration: 67, Log-Lik: -11599.335, Max-Change: 0.00011
Iteration: 68, Log-Lik: -11599.335, Max-Change: 0.00056
Iteration: 69, Log-Lik: -11599.335, Max-Change: 0.00019
Iteration: 70, Log-Lik: -11599.335, Max-Change: 0.00011
Iteration: 71, Log-Lik: -11599.335, Max-Change: 0.00055
Iteration: 72, Log-Lik: -11599.335, Max-Change: 0.00019
Iteration: 73, Log-Lik: -11599.335, Max-Change: 0.00011
Iteration: 74, Log-Lik: -11599.335, Max-Change: 0.00054
Iteration: 75, Log-Lik: -11599.335, Max-Change: 0.00018
Iteration: 76, Log-Lik: -11599.335, Max-Change: 0.00011
Iteration: 77, Log-Lik: -11599.335, Max-Change: 0.00052
Iteration: 78, Log-Lik: -11599.335, Max-Change: 0.00018
Iteration: 79, Log-Lik: -11599.335, Max-Change: 0.00010
Iteration: 80, Log-Lik: -11599.335, Max-Change: 0.00051
Iteration: 81, Log-Lik: -11599.335, Max-Change: 0.00017
Iteration: 82, Log-Lik: -11599.335, Max-Change: 0.00010
Iteration: 83, Log-Lik: -11599.335, Max-Change: 0.00050
Iteration: 84, Log-Lik: -11599.335, Max-Change: 0.00017
Iteration: 85, Log-Lik: -11599.335, Max-Change: 0.00010
coef(mod0)$Item.1
#>            a1         d g u
#> par 0.8107167 -1.042366 0 1
coef(mod1)$Item.1
#>             a1 ak0       ak1      ak2      ak3 ak4 d0         d1         d2
#> par -0.8772035   0 0.5286601 1.116593 1.129494   4  0 -0.1909232 0.01878861
#>             d3       d4
#> par -0.1258261 -5.65218
coef(mod2)$Item.1
#>            a1        d g u ak0        ak1        ak2       ak3 d0        d1
#> par 0.8102548 -1.04233 0 1   0 -0.5653287 -0.5712706 -3.025613  0 0.2117761
#>             d2        d3
#> par 0.06919723 -5.309272
itemplot(mod0, 1)

itemplot(mod1, 1)

itemplot(mod2, 1)


# compare added information from distractors
Theta <- matrix(seq(-4,4,.01))
par(mfrow = c(2,3))
for(i in 1:5){
    info <- iteminfo(extract.item(mod0,i), Theta)
    info2 <- iteminfo(extract.item(mod2,i), Theta)
    plot(Theta, info2, type = 'l', main = paste('Information for item', i), ylab = 'Information')
    lines(Theta, info, col = 'red')
}
par(mfrow = c(1,1))


# test information
plot(Theta, testinfo(mod2, Theta), type = 'l', main = 'Test information', ylab = 'Information')
lines(Theta, testinfo(mod0, Theta), col = 'red')


###########
# using the MH-RM algorithm
data(LSAT7)
fulldata <- expand.table(LSAT7)
(mod1 <- mirt(fulldata, 1, method = 'MHRM'))
#> 
Stage 1 = 1, CDLL = -3833.3, AR(2.68) = [0.49], Max-Change = 0.0343
Stage 1 = 2, CDLL = -3804.2, AR(2.68) = [0.49], Max-Change = 0.0321
Stage 1 = 3, CDLL = -3856.8, AR(2.68) = [0.49], Max-Change = 0.0244
Stage 1 = 4, CDLL = -3838.4, AR(2.68) = [0.46], Max-Change = 0.0454
Stage 1 = 5, CDLL = -3825.2, AR(2.68) = [0.46], Max-Change = 0.0518
Stage 1 = 6, CDLL = -3849.7, AR(2.68) = [0.47], Max-Change = 0.0217
Stage 1 = 7, CDLL = -3789.4, AR(2.68) = [0.46], Max-Change = 0.0328
Stage 1 = 8, CDLL = -3831.7, AR(2.68) = [0.52], Max-Change = 0.0227
Stage 1 = 9, CDLL = -3820.7, AR(2.68) = [0.50], Max-Change = 0.0287
Stage 1 = 10, CDLL = -3809.1, AR(2.68) = [0.48], Max-Change = 0.0194
Stage 1 = 11, CDLL = -3812.1, AR(2.68) = [0.48], Max-Change = 0.0210
Stage 1 = 12, CDLL = -3802.4, AR(2.68) = [0.47], Max-Change = 0.0517
Stage 1 = 13, CDLL = -3830.3, AR(2.68) = [0.48], Max-Change = 0.0631
Stage 1 = 14, CDLL = -3808.4, AR(2.68) = [0.46], Max-Change = 0.0216
Stage 1 = 15, CDLL = -3823.2, AR(2.68) = [0.47], Max-Change = 0.0277
Stage 1 = 16, CDLL = -3796.4, AR(2.68) = [0.46], Max-Change = 0.0191
Stage 1 = 17, CDLL = -3779.2, AR(2.68) = [0.47], Max-Change = 0.0253
Stage 1 = 18, CDLL = -3809.6, AR(2.68) = [0.48], Max-Change = 0.0380
Stage 1 = 19, CDLL = -3802.1, AR(2.68) = [0.47], Max-Change = 0.0219
Stage 1 = 20, CDLL = -3763.0, AR(2.68) = [0.44], Max-Change = 0.0382
Stage 1 = 21, CDLL = -3800.7, AR(2.68) = [0.46], Max-Change = 0.0325
Stage 1 = 22, CDLL = -3785.4, AR(2.68) = [0.47], Max-Change = 0.0429
Stage 1 = 23, CDLL = -3785.1, AR(2.68) = [0.50], Max-Change = 0.0375
Stage 1 = 24, CDLL = -3812.0, AR(2.68) = [0.45], Max-Change = 0.0238
Stage 1 = 25, CDLL = -3766.9, AR(2.68) = [0.46], Max-Change = 0.0391
Stage 1 = 26, CDLL = -3788.1, AR(2.68) = [0.48], Max-Change = 0.0400
Stage 1 = 27, CDLL = -3810.0, AR(2.68) = [0.50], Max-Change = 0.0139
Stage 1 = 28, CDLL = -3785.2, AR(2.68) = [0.48], Max-Change = 0.0325
Stage 1 = 29, CDLL = -3781.2, AR(2.68) = [0.45], Max-Change = 0.0143
Stage 1 = 30, CDLL = -3793.5, AR(2.68) = [0.49], Max-Change = 0.0271
Stage 1 = 31, CDLL = -3813.2, AR(2.68) = [0.48], Max-Change = 0.0202
Stage 1 = 32, CDLL = -3787.8, AR(2.68) = [0.48], Max-Change = 0.0155
Stage 1 = 33, CDLL = -3744.5, AR(2.68) = [0.47], Max-Change = 0.0353
Stage 1 = 34, CDLL = -3734.4, AR(2.68) = [0.45], Max-Change = 0.0603
Stage 1 = 35, CDLL = -3753.5, AR(2.68) = [0.46], Max-Change = 0.0193
Stage 1 = 36, CDLL = -3765.4, AR(2.68) = [0.47], Max-Change = 0.0270
Stage 1 = 37, CDLL = -3773.4, AR(2.68) = [0.49], Max-Change = 0.0234
Stage 1 = 38, CDLL = -3771.2, AR(2.68) = [0.47], Max-Change = 0.0329
Stage 1 = 39, CDLL = -3831.1, AR(2.68) = [0.47], Max-Change = 0.0450
Stage 1 = 40, CDLL = -3765.9, AR(2.68) = [0.46], Max-Change = 0.0438
Stage 1 = 41, CDLL = -3788.8, AR(2.68) = [0.47], Max-Change = 0.0420
Stage 1 = 42, CDLL = -3765.0, AR(2.68) = [0.47], Max-Change = 0.0558
Stage 1 = 43, CDLL = -3766.3, AR(2.68) = [0.46], Max-Change = 0.0300
Stage 1 = 44, CDLL = -3784.7, AR(2.68) = [0.49], Max-Change = 0.0203
Stage 1 = 45, CDLL = -3756.6, AR(2.68) = [0.45], Max-Change = 0.0384
Stage 1 = 46, CDLL = -3797.4, AR(2.68) = [0.47], Max-Change = 0.0183
Stage 1 = 47, CDLL = -3817.9, AR(2.68) = [0.50], Max-Change = 0.0178
Stage 1 = 48, CDLL = -3797.7, AR(2.68) = [0.45], Max-Change = 0.0142
Stage 1 = 49, CDLL = -3791.9, AR(2.68) = [0.47], Max-Change = 0.0165
Stage 1 = 50, CDLL = -3796.4, AR(2.68) = [0.45], Max-Change = 0.0152
Stage 1 = 51, CDLL = -3829.6, AR(2.68) = [0.47], Max-Change = 0.0430
Stage 1 = 52, CDLL = -3809.8, AR(2.68) = [0.47], Max-Change = 0.0577
Stage 1 = 53, CDLL = -3819.8, AR(2.68) = [0.45], Max-Change = 0.0273
Stage 1 = 54, CDLL = -3768.9, AR(2.68) = [0.47], Max-Change = 0.0351
Stage 1 = 55, CDLL = -3786.0, AR(2.68) = [0.49], Max-Change = 0.0191
Stage 1 = 56, CDLL = -3783.0, AR(2.68) = [0.47], Max-Change = 0.0106
Stage 1 = 57, CDLL = -3797.0, AR(2.68) = [0.44], Max-Change = 0.0242
Stage 1 = 58, CDLL = -3750.4, AR(2.68) = [0.49], Max-Change = 0.0290
Stage 1 = 59, CDLL = -3775.1, AR(2.68) = [0.46], Max-Change = 0.0189
Stage 1 = 60, CDLL = -3799.4, AR(2.68) = [0.48], Max-Change = 0.0211
Stage 1 = 61, CDLL = -3809.3, AR(2.68) = [0.49], Max-Change = 0.0176
Stage 1 = 62, CDLL = -3746.6, AR(2.68) = [0.45], Max-Change = 0.0322
Stage 1 = 63, CDLL = -3775.6, AR(2.68) = [0.47], Max-Change = 0.0263
Stage 1 = 64, CDLL = -3761.6, AR(2.68) = [0.49], Max-Change = 0.0259
Stage 1 = 65, CDLL = -3785.6, AR(2.68) = [0.45], Max-Change = 0.0295
Stage 1 = 66, CDLL = -3758.2, AR(2.68) = [0.47], Max-Change = 0.0401
Stage 1 = 67, CDLL = -3766.3, AR(2.68) = [0.44], Max-Change = 0.0123
Stage 1 = 68, CDLL = -3785.3, AR(2.68) = [0.49], Max-Change = 0.0411
Stage 1 = 69, CDLL = -3776.5, AR(2.68) = [0.49], Max-Change = 0.0175
Stage 1 = 70, CDLL = -3764.1, AR(2.68) = [0.46], Max-Change = 0.0356
Stage 1 = 71, CDLL = -3788.2, AR(2.68) = [0.48], Max-Change = 0.0190
Stage 1 = 72, CDLL = -3762.8, AR(2.68) = [0.45], Max-Change = 0.0254
Stage 1 = 73, CDLL = -3801.3, AR(2.68) = [0.46], Max-Change = 0.0190
Stage 1 = 74, CDLL = -3741.1, AR(2.68) = [0.47], Max-Change = 0.0235
Stage 1 = 75, CDLL = -3766.4, AR(2.68) = [0.45], Max-Change = 0.0216
Stage 1 = 76, CDLL = -3734.5, AR(2.68) = [0.44], Max-Change = 0.0183
Stage 1 = 77, CDLL = -3804.8, AR(2.68) = [0.47], Max-Change = 0.0151
Stage 1 = 78, CDLL = -3775.6, AR(2.68) = [0.46], Max-Change = 0.0392
Stage 1 = 79, CDLL = -3733.6, AR(2.68) = [0.46], Max-Change = 0.0207
Stage 1 = 80, CDLL = -3759.9, AR(2.68) = [0.45], Max-Change = 0.0135
Stage 1 = 81, CDLL = -3802.4, AR(2.68) = [0.47], Max-Change = 0.0314
Stage 1 = 82, CDLL = -3767.2, AR(2.68) = [0.46], Max-Change = 0.0380
Stage 1 = 83, CDLL = -3738.2, AR(2.68) = [0.44], Max-Change = 0.0442
Stage 1 = 84, CDLL = -3743.6, AR(2.68) = [0.48], Max-Change = 0.0163
Stage 1 = 85, CDLL = -3767.3, AR(2.68) = [0.46], Max-Change = 0.0227
Stage 1 = 86, CDLL = -3739.4, AR(2.68) = [0.44], Max-Change = 0.0268
Stage 1 = 87, CDLL = -3764.5, AR(2.68) = [0.49], Max-Change = 0.0140
Stage 1 = 88, CDLL = -3755.2, AR(2.68) = [0.46], Max-Change = 0.0211
Stage 1 = 89, CDLL = -3758.1, AR(2.68) = [0.48], Max-Change = 0.0146
Stage 1 = 90, CDLL = -3756.8, AR(2.68) = [0.44], Max-Change = 0.0419
Stage 1 = 91, CDLL = -3782.0, AR(2.68) = [0.46], Max-Change = 0.0299
Stage 1 = 92, CDLL = -3750.3, AR(2.68) = [0.46], Max-Change = 0.0168
Stage 1 = 93, CDLL = -3761.2, AR(2.68) = [0.45], Max-Change = 0.0297
Stage 1 = 94, CDLL = -3748.0, AR(2.68) = [0.46], Max-Change = 0.0207
Stage 1 = 95, CDLL = -3791.3, AR(2.68) = [0.47], Max-Change = 0.0229
Stage 1 = 96, CDLL = -3772.6, AR(2.68) = [0.47], Max-Change = 0.0329
Stage 1 = 97, CDLL = -3824.9, AR(2.68) = [0.45], Max-Change = 0.0384
Stage 1 = 98, CDLL = -3765.4, AR(2.68) = [0.44], Max-Change = 0.0203
Stage 1 = 99, CDLL = -3752.2, AR(2.68) = [0.42], Max-Change = 0.0273
Stage 1 = 100, CDLL = -3763.9, AR(2.68) = [0.45], Max-Change = 0.0129
Stage 1 = 101, CDLL = -3763.1, AR(2.68) = [0.45], Max-Change = 0.0143
Stage 1 = 102, CDLL = -3778.9, AR(2.68) = [0.48], Max-Change = 0.0248
Stage 1 = 103, CDLL = -3806.2, AR(2.68) = [0.48], Max-Change = 0.0268
Stage 1 = 104, CDLL = -3775.3, AR(2.68) = [0.45], Max-Change = 0.0267
Stage 1 = 105, CDLL = -3757.6, AR(2.68) = [0.45], Max-Change = 0.0342
Stage 1 = 106, CDLL = -3774.2, AR(2.68) = [0.46], Max-Change = 0.0256
Stage 1 = 107, CDLL = -3763.7, AR(2.68) = [0.48], Max-Change = 0.0346
Stage 1 = 108, CDLL = -3755.0, AR(2.68) = [0.48], Max-Change = 0.0357
Stage 1 = 109, CDLL = -3753.1, AR(2.68) = [0.48], Max-Change = 0.0225
Stage 1 = 110, CDLL = -3761.3, AR(2.68) = [0.44], Max-Change = 0.0236
Stage 1 = 111, CDLL = -3724.3, AR(2.68) = [0.46], Max-Change = 0.0390
Stage 1 = 112, CDLL = -3782.1, AR(2.68) = [0.47], Max-Change = 0.0243
Stage 1 = 113, CDLL = -3742.2, AR(2.68) = [0.49], Max-Change = 0.0328
Stage 1 = 114, CDLL = -3763.8, AR(2.68) = [0.47], Max-Change = 0.0294
Stage 1 = 115, CDLL = -3769.0, AR(2.68) = [0.44], Max-Change = 0.0478
Stage 1 = 116, CDLL = -3727.1, AR(2.68) = [0.46], Max-Change = 0.0269
Stage 1 = 117, CDLL = -3754.7, AR(2.68) = [0.48], Max-Change = 0.0292
Stage 1 = 118, CDLL = -3774.4, AR(2.68) = [0.47], Max-Change = 0.0329
Stage 1 = 119, CDLL = -3781.7, AR(2.68) = [0.47], Max-Change = 0.0127
Stage 1 = 120, CDLL = -3770.6, AR(2.68) = [0.45], Max-Change = 0.0280
Stage 1 = 121, CDLL = -3778.7, AR(2.68) = [0.48], Max-Change = 0.0355
Stage 1 = 122, CDLL = -3786.3, AR(2.68) = [0.46], Max-Change = 0.0357
Stage 1 = 123, CDLL = -3746.6, AR(2.68) = [0.45], Max-Change = 0.0183
Stage 1 = 124, CDLL = -3736.3, AR(2.68) = [0.45], Max-Change = 0.0201
Stage 1 = 125, CDLL = -3773.6, AR(2.68) = [0.49], Max-Change = 0.0258
Stage 1 = 126, CDLL = -3765.3, AR(2.68) = [0.45], Max-Change = 0.0241
Stage 1 = 127, CDLL = -3774.4, AR(2.68) = [0.47], Max-Change = 0.0265
Stage 1 = 128, CDLL = -3727.8, AR(2.68) = [0.45], Max-Change = 0.0413
Stage 1 = 129, CDLL = -3713.2, AR(2.68) = [0.47], Max-Change = 0.0309
Stage 1 = 130, CDLL = -3707.5, AR(2.68) = [0.46], Max-Change = 0.0368
Stage 1 = 131, CDLL = -3719.3, AR(2.68) = [0.47], Max-Change = 0.0182
Stage 1 = 132, CDLL = -3738.7, AR(2.68) = [0.45], Max-Change = 0.0521
Stage 1 = 133, CDLL = -3729.5, AR(2.68) = [0.47], Max-Change = 0.0192
Stage 1 = 134, CDLL = -3713.2, AR(2.68) = [0.45], Max-Change = 0.0279
Stage 1 = 135, CDLL = -3769.9, AR(2.68) = [0.48], Max-Change = 0.0287
Stage 1 = 136, CDLL = -3739.5, AR(2.68) = [0.44], Max-Change = 0.0111
Stage 1 = 137, CDLL = -3743.4, AR(2.68) = [0.46], Max-Change = 0.0109
Stage 1 = 138, CDLL = -3710.5, AR(2.68) = [0.45], Max-Change = 0.0181
Stage 1 = 139, CDLL = -3713.8, AR(2.68) = [0.44], Max-Change = 0.0334
Stage 1 = 140, CDLL = -3770.8, AR(2.68) = [0.46], Max-Change = 0.0324
Stage 1 = 141, CDLL = -3778.2, AR(2.68) = [0.48], Max-Change = 0.0434
Stage 1 = 142, CDLL = -3769.0, AR(2.68) = [0.47], Max-Change = 0.0336
Stage 1 = 143, CDLL = -3739.5, AR(2.68) = [0.45], Max-Change = 0.0361
Stage 1 = 144, CDLL = -3741.4, AR(2.68) = [0.47], Max-Change = 0.0365
Stage 1 = 145, CDLL = -3776.6, AR(2.68) = [0.45], Max-Change = 0.0401
Stage 1 = 146, CDLL = -3765.2, AR(2.68) = [0.44], Max-Change = 0.0390
Stage 1 = 147, CDLL = -3760.9, AR(2.68) = [0.48], Max-Change = 0.0134
Stage 1 = 148, CDLL = -3790.4, AR(2.68) = [0.46], Max-Change = 0.0310
Stage 1 = 149, CDLL = -3792.6, AR(2.68) = [0.47], Max-Change = 0.0252
Stage 1 = 150, CDLL = -3780.3, AR(4.07) = [0.40], Max-Change = 0.0230
Stage 2 = 1, CDLL = -3777.1, AR(4.07) = [0.38], Max-Change = 0.0128
Stage 2 = 2, CDLL = -3757.8, AR(4.07) = [0.41], Max-Change = 0.0381
Stage 2 = 3, CDLL = -3788.9, AR(4.07) = [0.39], Max-Change = 0.0210
Stage 2 = 4, CDLL = -3774.1, AR(4.07) = [0.41], Max-Change = 0.0256
Stage 2 = 5, CDLL = -3752.6, AR(4.07) = [0.37], Max-Change = 0.0320
Stage 2 = 6, CDLL = -3768.7, AR(4.07) = [0.38], Max-Change = 0.0227
Stage 2 = 7, CDLL = -3765.5, AR(4.07) = [0.40], Max-Change = 0.0332
Stage 2 = 8, CDLL = -3751.2, AR(4.07) = [0.39], Max-Change = 0.0103
Stage 2 = 9, CDLL = -3763.4, AR(4.07) = [0.39], Max-Change = 0.0267
Stage 2 = 10, CDLL = -3768.6, AR(4.07) = [0.40], Max-Change = 0.0244
Stage 2 = 11, CDLL = -3724.7, AR(4.07) = [0.41], Max-Change = 0.0244
Stage 2 = 12, CDLL = -3723.0, AR(4.07) = [0.39], Max-Change = 0.0336
Stage 2 = 13, CDLL = -3744.9, AR(4.07) = [0.39], Max-Change = 0.0209
Stage 2 = 14, CDLL = -3754.4, AR(4.07) = [0.38], Max-Change = 0.0287
Stage 2 = 15, CDLL = -3734.6, AR(4.07) = [0.39], Max-Change = 0.0210
Stage 2 = 16, CDLL = -3756.5, AR(4.07) = [0.40], Max-Change = 0.0163
Stage 2 = 17, CDLL = -3801.6, AR(4.07) = [0.38], Max-Change = 0.0394
Stage 2 = 18, CDLL = -3765.6, AR(4.07) = [0.39], Max-Change = 0.0194
Stage 2 = 19, CDLL = -3732.1, AR(4.07) = [0.40], Max-Change = 0.0141
Stage 2 = 20, CDLL = -3768.1, AR(4.07) = [0.39], Max-Change = 0.0347
Stage 2 = 21, CDLL = -3779.8, AR(4.07) = [0.42], Max-Change = 0.0188
Stage 2 = 22, CDLL = -3783.8, AR(4.07) = [0.37], Max-Change = 0.0249
Stage 2 = 23, CDLL = -3807.3, AR(4.07) = [0.40], Max-Change = 0.0504
Stage 2 = 24, CDLL = -3767.7, AR(4.07) = [0.38], Max-Change = 0.0254
Stage 2 = 25, CDLL = -3763.7, AR(4.07) = [0.40], Max-Change = 0.0192
Stage 2 = 26, CDLL = -3770.7, AR(4.07) = [0.39], Max-Change = 0.0197
Stage 2 = 27, CDLL = -3785.8, AR(4.07) = [0.41], Max-Change = 0.0477
Stage 2 = 28, CDLL = -3783.5, AR(4.07) = [0.39], Max-Change = 0.0258
Stage 2 = 29, CDLL = -3770.1, AR(4.07) = [0.38], Max-Change = 0.0296
Stage 2 = 30, CDLL = -3765.3, AR(4.07) = [0.43], Max-Change = 0.0425
Stage 2 = 31, CDLL = -3754.7, AR(4.07) = [0.38], Max-Change = 0.0353
Stage 2 = 32, CDLL = -3738.8, AR(4.07) = [0.40], Max-Change = 0.0328
Stage 2 = 33, CDLL = -3783.6, AR(4.07) = [0.39], Max-Change = 0.0186
Stage 2 = 34, CDLL = -3782.7, AR(4.07) = [0.39], Max-Change = 0.0190
Stage 2 = 35, CDLL = -3803.3, AR(4.07) = [0.40], Max-Change = 0.0243
Stage 2 = 36, CDLL = -3806.7, AR(4.07) = [0.39], Max-Change = 0.0284
Stage 2 = 37, CDLL = -3757.5, AR(4.07) = [0.40], Max-Change = 0.0441
Stage 2 = 38, CDLL = -3779.1, AR(4.07) = [0.42], Max-Change = 0.0193
Stage 2 = 39, CDLL = -3758.4, AR(4.07) = [0.42], Max-Change = 0.0185
Stage 2 = 40, CDLL = -3791.2, AR(4.07) = [0.41], Max-Change = 0.0279
Stage 2 = 41, CDLL = -3759.9, AR(4.07) = [0.40], Max-Change = 0.0206
Stage 2 = 42, CDLL = -3768.6, AR(4.07) = [0.41], Max-Change = 0.0270
Stage 2 = 43, CDLL = -3807.4, AR(4.07) = [0.41], Max-Change = 0.0225
Stage 2 = 44, CDLL = -3782.4, AR(4.07) = [0.38], Max-Change = 0.0261
Stage 2 = 45, CDLL = -3766.6, AR(4.07) = [0.41], Max-Change = 0.0200
Stage 2 = 46, CDLL = -3771.6, AR(4.07) = [0.40], Max-Change = 0.0242
Stage 2 = 47, CDLL = -3807.9, AR(4.07) = [0.39], Max-Change = 0.0229
Stage 2 = 48, CDLL = -3774.0, AR(4.07) = [0.38], Max-Change = 0.0176
Stage 2 = 49, CDLL = -3797.4, AR(4.07) = [0.41], Max-Change = 0.0300
Stage 2 = 50, CDLL = -3787.3, AR(4.07) = [0.41], Max-Change = 0.0435
Stage 2 = 51, CDLL = -3798.5, AR(4.07) = [0.40], Max-Change = 0.0193
Stage 2 = 52, CDLL = -3737.7, AR(4.07) = [0.42], Max-Change = 0.0254
Stage 2 = 53, CDLL = -3788.5, AR(4.07) = [0.39], Max-Change = 0.0431
Stage 2 = 54, CDLL = -3779.4, AR(4.07) = [0.39], Max-Change = 0.0187
Stage 2 = 55, CDLL = -3808.0, AR(4.07) = [0.39], Max-Change = 0.0259
Stage 2 = 56, CDLL = -3805.4, AR(4.07) = [0.38], Max-Change = 0.0235
Stage 2 = 57, CDLL = -3811.6, AR(4.07) = [0.41], Max-Change = 0.0274
Stage 2 = 58, CDLL = -3789.5, AR(4.07) = [0.41], Max-Change = 0.0291
Stage 2 = 59, CDLL = -3810.7, AR(4.07) = [0.41], Max-Change = 0.0226
Stage 2 = 60, CDLL = -3764.1, AR(4.07) = [0.41], Max-Change = 0.0347
Stage 2 = 61, CDLL = -3773.3, AR(4.07) = [0.42], Max-Change = 0.0352
Stage 2 = 62, CDLL = -3768.1, AR(4.07) = [0.40], Max-Change = 0.0129
Stage 2 = 63, CDLL = -3815.9, AR(4.07) = [0.41], Max-Change = 0.0599
Stage 2 = 64, CDLL = -3822.5, AR(4.07) = [0.40], Max-Change = 0.0234
Stage 2 = 65, CDLL = -3734.5, AR(4.07) = [0.43], Max-Change = 0.0325
Stage 2 = 66, CDLL = -3780.4, AR(4.07) = [0.41], Max-Change = 0.0117
Stage 2 = 67, CDLL = -3746.6, AR(4.07) = [0.41], Max-Change = 0.0211
Stage 2 = 68, CDLL = -3800.7, AR(4.07) = [0.41], Max-Change = 0.0094
Stage 2 = 69, CDLL = -3743.0, AR(4.07) = [0.43], Max-Change = 0.0331
Stage 2 = 70, CDLL = -3736.9, AR(4.07) = [0.40], Max-Change = 0.0236
Stage 2 = 71, CDLL = -3740.0, AR(4.07) = [0.38], Max-Change = 0.0391
Stage 2 = 72, CDLL = -3725.1, AR(4.07) = [0.43], Max-Change = 0.0287
Stage 2 = 73, CDLL = -3769.4, AR(4.07) = [0.40], Max-Change = 0.0359
Stage 2 = 74, CDLL = -3794.0, AR(4.07) = [0.42], Max-Change = 0.0266
Stage 2 = 75, CDLL = -3793.8, AR(4.07) = [0.40], Max-Change = 0.0286
Stage 2 = 76, CDLL = -3774.9, AR(4.07) = [0.37], Max-Change = 0.0188
Stage 2 = 77, CDLL = -3730.3, AR(4.07) = [0.41], Max-Change = 0.0268
Stage 2 = 78, CDLL = -3769.7, AR(4.07) = [0.40], Max-Change = 0.0187
Stage 2 = 79, CDLL = -3796.4, AR(4.07) = [0.41], Max-Change = 0.0217
Stage 2 = 80, CDLL = -3779.3, AR(4.07) = [0.39], Max-Change = 0.0414
Stage 2 = 81, CDLL = -3770.7, AR(4.07) = [0.38], Max-Change = 0.0215
Stage 2 = 82, CDLL = -3788.1, AR(4.07) = [0.39], Max-Change = 0.0299
Stage 2 = 83, CDLL = -3779.4, AR(4.07) = [0.41], Max-Change = 0.0270
Stage 2 = 84, CDLL = -3717.7, AR(4.07) = [0.39], Max-Change = 0.0305
Stage 2 = 85, CDLL = -3708.7, AR(4.07) = [0.39], Max-Change = 0.0661
Stage 2 = 86, CDLL = -3754.0, AR(4.07) = [0.41], Max-Change = 0.0131
Stage 2 = 87, CDLL = -3742.8, AR(4.07) = [0.37], Max-Change = 0.0423
Stage 2 = 88, CDLL = -3734.7, AR(4.07) = [0.38], Max-Change = 0.0174
Stage 2 = 89, CDLL = -3765.9, AR(4.07) = [0.39], Max-Change = 0.0311
Stage 2 = 90, CDLL = -3724.8, AR(4.07) = [0.36], Max-Change = 0.0227
Stage 2 = 91, CDLL = -3743.1, AR(4.07) = [0.39], Max-Change = 0.0335
Stage 2 = 92, CDLL = -3745.1, AR(4.07) = [0.40], Max-Change = 0.0233
Stage 2 = 93, CDLL = -3752.0, AR(4.07) = [0.44], Max-Change = 0.0172
Stage 2 = 94, CDLL = -3756.2, AR(4.07) = [0.39], Max-Change = 0.0584
Stage 2 = 95, CDLL = -3766.7, AR(4.07) = [0.41], Max-Change = 0.0442
Stage 2 = 96, CDLL = -3745.4, AR(4.07) = [0.40], Max-Change = 0.0101
Stage 2 = 97, CDLL = -3767.8, AR(4.07) = [0.39], Max-Change = 0.0284
Stage 2 = 98, CDLL = -3795.8, AR(4.07) = [0.39], Max-Change = 0.0279
Stage 2 = 99, CDLL = -3754.4, AR(4.07) = [0.41], Max-Change = 0.0353
Stage 2 = 100, CDLL = -3766.2, AR(4.07) = [0.42], Max-Change = 0.0325
Stage 3 = 1, CDLL = -3823.8, AR(4.07) = [0.41], gam = 0.0000, Max-Change = 0.0000
Stage 3 = 2, CDLL = -3775.3, AR(4.07) = [0.42], gam = 0.1778, Max-Change = 0.0136
Stage 3 = 3, CDLL = -3776.0, AR(4.07) = [0.40], gam = 0.1057, Max-Change = 0.0187
Stage 3 = 4, CDLL = -3756.9, AR(4.07) = [0.41], gam = 0.0780, Max-Change = 0.0054
Stage 3 = 5, CDLL = -3763.3, AR(4.07) = [0.40], gam = 0.0629, Max-Change = 0.0068
Stage 3 = 6, CDLL = -3767.0, AR(4.07) = [0.42], gam = 0.0532, Max-Change = 0.0054
Stage 3 = 7, CDLL = -3778.9, AR(4.07) = [0.39], gam = 0.0464, Max-Change = 0.0051
Stage 3 = 8, CDLL = -3765.0, AR(4.07) = [0.43], gam = 0.0413, Max-Change = 0.0079
Stage 3 = 9, CDLL = -3790.8, AR(4.07) = [0.42], gam = 0.0374, Max-Change = 0.0091
Stage 3 = 10, CDLL = -3777.9, AR(4.07) = [0.43], gam = 0.0342, Max-Change = 0.0035
Stage 3 = 11, CDLL = -3757.8, AR(4.07) = [0.40], gam = 0.0316, Max-Change = 0.0049
Stage 3 = 12, CDLL = -3747.3, AR(4.07) = [0.39], gam = 0.0294, Max-Change = 0.0015
Stage 3 = 13, CDLL = -3783.9, AR(4.07) = [0.41], gam = 0.0276, Max-Change = 0.0042
Stage 3 = 14, CDLL = -3760.3, AR(4.07) = [0.41], gam = 0.0260, Max-Change = 0.0040
Stage 3 = 15, CDLL = -3734.2, AR(4.07) = [0.41], gam = 0.0246, Max-Change = 0.0085
Stage 3 = 16, CDLL = -3777.1, AR(4.07) = [0.41], gam = 0.0233, Max-Change = 0.0034
Stage 3 = 17, CDLL = -3750.2, AR(4.07) = [0.42], gam = 0.0222, Max-Change = 0.0031
Stage 3 = 18, CDLL = -3785.3, AR(4.07) = [0.41], gam = 0.0212, Max-Change = 0.0021
Stage 3 = 19, CDLL = -3771.8, AR(4.07) = [0.40], gam = 0.0203, Max-Change = 0.0027
Stage 3 = 20, CDLL = -3767.6, AR(4.07) = [0.37], gam = 0.0195, Max-Change = 0.0016
Stage 3 = 21, CDLL = -3753.3, AR(4.07) = [0.42], gam = 0.0188, Max-Change = 0.0026
Stage 3 = 22, CDLL = -3748.1, AR(4.07) = [0.40], gam = 0.0181, Max-Change = 0.0033
Stage 3 = 23, CDLL = -3740.7, AR(4.07) = [0.40], gam = 0.0175, Max-Change = 0.0036
Stage 3 = 24, CDLL = -3729.5, AR(4.07) = [0.37], gam = 0.0169, Max-Change = 0.0023
Stage 3 = 25, CDLL = -3771.1, AR(4.07) = [0.41], gam = 0.0164, Max-Change = 0.0019
Stage 3 = 26, CDLL = -3772.2, AR(4.07) = [0.43], gam = 0.0159, Max-Change = 0.0020
Stage 3 = 27, CDLL = -3742.7, AR(4.07) = [0.40], gam = 0.0154, Max-Change = 0.0024
Stage 3 = 28, CDLL = -3736.1, AR(4.07) = [0.41], gam = 0.0150, Max-Change = 0.0012
Stage 3 = 29, CDLL = -3771.7, AR(4.07) = [0.42], gam = 0.0146, Max-Change = 0.0021
Stage 3 = 30, CDLL = -3757.6, AR(4.07) = [0.39], gam = 0.0142, Max-Change = 0.0013
Stage 3 = 31, CDLL = -3767.4, AR(4.07) = [0.39], gam = 0.0139, Max-Change = 0.0034
Stage 3 = 32, CDLL = -3792.0, AR(4.07) = [0.41], gam = 0.0135, Max-Change = 0.0010
Stage 3 = 33, CDLL = -3761.3, AR(4.07) = [0.41], gam = 0.0132, Max-Change = 0.0017
Stage 3 = 34, CDLL = -3720.3, AR(4.07) = [0.40], gam = 0.0129, Max-Change = 0.0029
Stage 3 = 35, CDLL = -3760.0, AR(4.07) = [0.40], gam = 0.0126, Max-Change = 0.0019
Stage 3 = 36, CDLL = -3786.2, AR(4.07) = [0.40], gam = 0.0124, Max-Change = 0.0015
Stage 3 = 37, CDLL = -3787.9, AR(4.07) = [0.39], gam = 0.0121, Max-Change = 0.0030
Stage 3 = 38, CDLL = -3757.0, AR(4.07) = [0.38], gam = 0.0119, Max-Change = 0.0017
Stage 3 = 39, CDLL = -3751.0, AR(4.07) = [0.40], gam = 0.0116, Max-Change = 0.0006
Stage 3 = 40, CDLL = -3765.0, AR(4.07) = [0.39], gam = 0.0114, Max-Change = 0.0007
Stage 3 = 41, CDLL = -3759.6, AR(4.07) = [0.37], gam = 0.0112, Max-Change = 0.0013
Stage 3 = 42, CDLL = -3779.0, AR(4.07) = [0.37], gam = 0.0110, Max-Change = 0.0008
Stage 3 = 43, CDLL = -3784.7, AR(4.07) = [0.39], gam = 0.0108, Max-Change = 0.0011
Stage 3 = 44, CDLL = -3766.9, AR(4.07) = [0.40], gam = 0.0106, Max-Change = 0.0010
Stage 3 = 45, CDLL = -3776.8, AR(4.07) = [0.39], gam = 0.0104, Max-Change = 0.0014
Stage 3 = 46, CDLL = -3738.1, AR(4.07) = [0.41], gam = 0.0102, Max-Change = 0.0023
Stage 3 = 47, CDLL = -3739.4, AR(4.07) = [0.38], gam = 0.0101, Max-Change = 0.0025
Stage 3 = 48, CDLL = -3731.5, AR(4.07) = [0.38], gam = 0.0099, Max-Change = 0.0013
Stage 3 = 49, CDLL = -3729.7, AR(4.07) = [0.39], gam = 0.0098, Max-Change = 0.0014
Stage 3 = 50, CDLL = -3783.6, AR(4.07) = [0.41], gam = 0.0096, Max-Change = 0.0018
Stage 3 = 51, CDLL = -3766.0, AR(4.07) = [0.38], gam = 0.0095, Max-Change = 0.0012
Stage 3 = 52, CDLL = -3761.5, AR(4.07) = [0.41], gam = 0.0093, Max-Change = 0.0008
Stage 3 = 53, CDLL = -3761.2, AR(4.07) = [0.39], gam = 0.0092, Max-Change = 0.0009
Stage 3 = 54, CDLL = -3731.9, AR(4.07) = [0.42], gam = 0.0091, Max-Change = 0.0019
Stage 3 = 55, CDLL = -3722.5, AR(4.07) = [0.41], gam = 0.0089, Max-Change = 0.0030
Stage 3 = 56, CDLL = -3738.2, AR(4.07) = [0.39], gam = 0.0088, Max-Change = 0.0014
Stage 3 = 57, CDLL = -3754.7, AR(4.07) = [0.39], gam = 0.0087, Max-Change = 0.0004
Stage 3 = 58, CDLL = -3723.0, AR(4.07) = [0.41], gam = 0.0086, Max-Change = 0.0010
Stage 3 = 59, CDLL = -3739.4, AR(4.07) = [0.40], gam = 0.0085, Max-Change = 0.0009
Stage 3 = 60, CDLL = -3794.1, AR(4.07) = [0.39], gam = 0.0084, Max-Change = 0.0023
Stage 3 = 61, CDLL = -3769.9, AR(4.07) = [0.40], gam = 0.0082, Max-Change = 0.0013
Stage 3 = 62, CDLL = -3795.6, AR(4.07) = [0.39], gam = 0.0081, Max-Change = 0.0007
Stage 3 = 63, CDLL = -3780.4, AR(4.07) = [0.41], gam = 0.0080, Max-Change = 0.0006
Stage 3 = 64, CDLL = -3788.7, AR(4.07) = [0.40], gam = 0.0080, Max-Change = 0.0015
Stage 3 = 65, CDLL = -3801.5, AR(4.07) = [0.39], gam = 0.0079, Max-Change = 0.0006
Stage 3 = 66, CDLL = -3723.5, AR(4.07) = [0.40], gam = 0.0078, Max-Change = 0.0023
Stage 3 = 67, CDLL = -3766.9, AR(4.07) = [0.43], gam = 0.0077, Max-Change = 0.0007
Stage 3 = 68, CDLL = -3743.6, AR(4.07) = [0.41], gam = 0.0076, Max-Change = 0.0003
Stage 3 = 69, CDLL = -3779.0, AR(4.07) = [0.40], gam = 0.0075, Max-Change = 0.0014
Stage 3 = 70, CDLL = -3778.1, AR(4.07) = [0.40], gam = 0.0074, Max-Change = 0.0012
Stage 3 = 71, CDLL = -3772.4, AR(4.07) = [0.37], gam = 0.0073, Max-Change = 0.0009
Stage 3 = 72, CDLL = -3747.9, AR(4.07) = [0.41], gam = 0.0073, Max-Change = 0.0008
Stage 3 = 73, CDLL = -3779.6, AR(4.07) = [0.39], gam = 0.0072, Max-Change = 0.0006

#> 
#> Calculating log-likelihood...
#> 
#> Call:
#> mirt(data = fulldata, model = 1, method = "MHRM")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 0.001 tolerance after 73 MHRM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: NR1 
#> Latent density type: Gaussian 
#> Average MH acceptance ratio(s): 0.4 
#> 
#> Log-likelihood = -2659.472, SE = 0.018
#> Estimated parameters: 10 
#> AIC = 5338.944
#> BIC = 5388.022; SABIC = 5356.261
#> G2 (21) = 32.89, p = 0.0475
#> RMSEA = 0.024, CFI = NaN, TLI = NaN

# Confirmatory models

# simulate data
a <- matrix(c(
1.5,NA,
0.5,NA,
1.0,NA,
1.0,0.5,
 NA,1.5,
 NA,0.5,
 NA,1.0,
 NA,1.0),ncol=2,byrow=TRUE)

d <- matrix(c(
-1.0,NA,NA,
-1.5,NA,NA,
 1.5,NA,NA,
 0.0,NA,NA,
3.0,2.0,-0.5,
2.5,1.0,-1,
2.0,0.0,NA,
1.0,NA,NA),ncol=3,byrow=TRUE)

sigma <- diag(2)
sigma[1,2] <- sigma[2,1] <- .4
items <- c(rep('2PL',4), rep('graded',3), '2PL')
dataset <- simdata(a,d,2000,items,sigma)

# analyses
# CIFA for 2 factor crossed structure

model.1 <- '
  F1 = 1-4
  F2 = 4-8
  COV = F1*F2'

# compute model, and use parallel computation of the log-likelihood
if(interactive()) mirtCluster()
mod1 <- mirt(dataset, model.1, method = 'MHRM')
#> 
Stage 1 = 1, CDLL = -17059.4, AR(0.77) = [0.49], Max-Change = 0.0353
Stage 1 = 2, CDLL = -17078.8, AR(0.77) = [0.47], Max-Change = 0.0297
Stage 1 = 3, CDLL = -17054.3, AR(0.77) = [0.49], Max-Change = 0.0175
Stage 1 = 4, CDLL = -17047.6, AR(0.77) = [0.49], Max-Change = 0.0285
Stage 1 = 5, CDLL = -16983.7, AR(0.77) = [0.50], Max-Change = 0.0349
Stage 1 = 6, CDLL = -16979.0, AR(0.77) = [0.50], Max-Change = 0.0324
Stage 1 = 7, CDLL = -16926.1, AR(0.77) = [0.49], Max-Change = 0.0255
Stage 1 = 8, CDLL = -16973.2, AR(0.77) = [0.50], Max-Change = 0.0210
Stage 1 = 9, CDLL = -16940.7, AR(0.77) = [0.48], Max-Change = 0.0321
Stage 1 = 10, CDLL = -16977.3, AR(0.77) = [0.48], Max-Change = 0.0269
Stage 1 = 11, CDLL = -17006.5, AR(0.77) = [0.51], Max-Change = 0.0250
Stage 1 = 12, CDLL = -16982.9, AR(0.77) = [0.48], Max-Change = 0.0204
Stage 1 = 13, CDLL = -16891.5, AR(0.77) = [0.49], Max-Change = 0.0280
Stage 1 = 14, CDLL = -16923.4, AR(0.77) = [0.47], Max-Change = 0.0252
Stage 1 = 15, CDLL = -16927.3, AR(0.77) = [0.50], Max-Change = 0.0169
Stage 1 = 16, CDLL = -16887.9, AR(0.77) = [0.47], Max-Change = 0.0363
Stage 1 = 17, CDLL = -16851.3, AR(0.77) = [0.48], Max-Change = 0.0308
Stage 1 = 18, CDLL = -16893.0, AR(0.77) = [0.47], Max-Change = 0.0254
Stage 1 = 19, CDLL = -16799.5, AR(0.77) = [0.50], Max-Change = 0.0259
Stage 1 = 20, CDLL = -16786.3, AR(0.77) = [0.48], Max-Change = 0.0185
Stage 1 = 21, CDLL = -16851.8, AR(0.77) = [0.49], Max-Change = 0.0172
Stage 1 = 22, CDLL = -16810.6, AR(0.77) = [0.48], Max-Change = 0.0221
Stage 1 = 23, CDLL = -16788.4, AR(0.77) = [0.46], Max-Change = 0.0230
Stage 1 = 24, CDLL = -16769.9, AR(0.77) = [0.48], Max-Change = 0.0283
Stage 1 = 25, CDLL = -16726.3, AR(0.77) = [0.48], Max-Change = 0.0209
Stage 1 = 26, CDLL = -16787.8, AR(0.77) = [0.48], Max-Change = 0.0271
Stage 1 = 27, CDLL = -16836.4, AR(0.77) = [0.50], Max-Change = 0.0141
Stage 1 = 28, CDLL = -16721.3, AR(0.77) = [0.50], Max-Change = 0.0247
Stage 1 = 29, CDLL = -16765.6, AR(0.77) = [0.47], Max-Change = 0.0146
Stage 1 = 30, CDLL = -16782.0, AR(0.77) = [0.48], Max-Change = 0.0126
Stage 1 = 31, CDLL = -16832.0, AR(0.77) = [0.48], Max-Change = 0.0282
Stage 1 = 32, CDLL = -16869.3, AR(0.77) = [0.49], Max-Change = 0.0124
Stage 1 = 33, CDLL = -16792.1, AR(0.77) = [0.48], Max-Change = 0.0136
Stage 1 = 34, CDLL = -16692.6, AR(0.77) = [0.48], Max-Change = 0.0407
Stage 1 = 35, CDLL = -16747.9, AR(0.77) = [0.50], Max-Change = 0.0132
Stage 1 = 36, CDLL = -16721.9, AR(0.77) = [0.48], Max-Change = 0.0167
Stage 1 = 37, CDLL = -16703.9, AR(0.77) = [0.47], Max-Change = 0.0267
Stage 1 = 38, CDLL = -16699.7, AR(0.77) = [0.47], Max-Change = 0.0289
Stage 1 = 39, CDLL = -16750.8, AR(0.77) = [0.46], Max-Change = 0.0163
Stage 1 = 40, CDLL = -16693.4, AR(0.77) = [0.46], Max-Change = 0.0121
Stage 1 = 41, CDLL = -16726.9, AR(0.77) = [0.48], Max-Change = 0.0121
Stage 1 = 42, CDLL = -16720.5, AR(0.77) = [0.47], Max-Change = 0.0181
Stage 1 = 43, CDLL = -16777.2, AR(0.77) = [0.49], Max-Change = 0.0219
Stage 1 = 44, CDLL = -16854.4, AR(0.77) = [0.47], Max-Change = 0.0277
Stage 1 = 45, CDLL = -16782.5, AR(0.77) = [0.48], Max-Change = 0.0213
Stage 1 = 46, CDLL = -16829.9, AR(0.77) = [0.47], Max-Change = 0.0295
Stage 1 = 47, CDLL = -16745.8, AR(0.77) = [0.45], Max-Change = 0.0187
Stage 1 = 48, CDLL = -16773.3, AR(0.77) = [0.48], Max-Change = 0.0171
Stage 1 = 49, CDLL = -16765.8, AR(0.77) = [0.47], Max-Change = 0.0181
Stage 1 = 50, CDLL = -16718.5, AR(0.77) = [0.48], Max-Change = 0.0217
Stage 1 = 51, CDLL = -16824.7, AR(0.77) = [0.48], Max-Change = 0.0242
Stage 1 = 52, CDLL = -16734.8, AR(0.77) = [0.46], Max-Change = 0.0089
Stage 1 = 53, CDLL = -16726.1, AR(0.77) = [0.47], Max-Change = 0.0132
Stage 1 = 54, CDLL = -16726.6, AR(0.77) = [0.44], Max-Change = 0.0082
Stage 1 = 55, CDLL = -16737.1, AR(0.77) = [0.47], Max-Change = 0.0152
Stage 1 = 56, CDLL = -16710.7, AR(0.77) = [0.45], Max-Change = 0.0143
Stage 1 = 57, CDLL = -16713.0, AR(0.77) = [0.47], Max-Change = 0.0076
Stage 1 = 58, CDLL = -16724.4, AR(0.77) = [0.48], Max-Change = 0.0186
Stage 1 = 59, CDLL = -16812.3, AR(0.77) = [0.47], Max-Change = 0.0204
Stage 1 = 60, CDLL = -16827.2, AR(0.77) = [0.47], Max-Change = 0.0286
Stage 1 = 61, CDLL = -16721.9, AR(0.77) = [0.47], Max-Change = 0.0179
Stage 1 = 62, CDLL = -16740.7, AR(0.77) = [0.46], Max-Change = 0.0149
Stage 1 = 63, CDLL = -16775.7, AR(0.77) = [0.46], Max-Change = 0.0103
Stage 1 = 64, CDLL = -16702.4, AR(0.77) = [0.46], Max-Change = 0.0187
Stage 1 = 65, CDLL = -16800.7, AR(0.77) = [0.46], Max-Change = 0.0158
Stage 1 = 66, CDLL = -16681.5, AR(0.77) = [0.46], Max-Change = 0.0174
Stage 1 = 67, CDLL = -16751.4, AR(0.77) = [0.46], Max-Change = 0.0210
Stage 1 = 68, CDLL = -16720.5, AR(0.77) = [0.50], Max-Change = 0.0153
Stage 1 = 69, CDLL = -16724.7, AR(0.77) = [0.47], Max-Change = 0.0141
Stage 1 = 70, CDLL = -16659.9, AR(0.77) = [0.46], Max-Change = 0.0152
Stage 1 = 71, CDLL = -16688.4, AR(0.77) = [0.48], Max-Change = 0.0139
Stage 1 = 72, CDLL = -16699.5, AR(0.77) = [0.47], Max-Change = 0.0212
Stage 1 = 73, CDLL = -16766.8, AR(0.77) = [0.48], Max-Change = 0.0161
Stage 1 = 74, CDLL = -16690.7, AR(0.77) = [0.47], Max-Change = 0.0162
Stage 1 = 75, CDLL = -16691.2, AR(0.77) = [0.48], Max-Change = 0.0151
Stage 1 = 76, CDLL = -16694.8, AR(0.77) = [0.47], Max-Change = 0.0178
Stage 1 = 77, CDLL = -16746.3, AR(0.77) = [0.46], Max-Change = 0.0126
Stage 1 = 78, CDLL = -16747.3, AR(0.77) = [0.47], Max-Change = 0.0200
Stage 1 = 79, CDLL = -16717.6, AR(0.77) = [0.47], Max-Change = 0.0176
Stage 1 = 80, CDLL = -16660.9, AR(0.77) = [0.47], Max-Change = 0.0282
Stage 1 = 81, CDLL = -16722.2, AR(0.77) = [0.49], Max-Change = 0.0166
Stage 1 = 82, CDLL = -16686.4, AR(0.77) = [0.46], Max-Change = 0.0196
Stage 1 = 83, CDLL = -16748.3, AR(0.77) = [0.47], Max-Change = 0.0191
Stage 1 = 84, CDLL = -16709.5, AR(0.77) = [0.46], Max-Change = 0.0270
Stage 1 = 85, CDLL = -16720.4, AR(0.77) = [0.47], Max-Change = 0.0214
Stage 1 = 86, CDLL = -16718.1, AR(0.77) = [0.46], Max-Change = 0.0196
Stage 1 = 87, CDLL = -16670.7, AR(0.77) = [0.45], Max-Change = 0.0182
Stage 1 = 88, CDLL = -16712.0, AR(0.77) = [0.47], Max-Change = 0.0120
Stage 1 = 89, CDLL = -16728.5, AR(0.77) = [0.46], Max-Change = 0.0228
Stage 1 = 90, CDLL = -16794.3, AR(0.77) = [0.47], Max-Change = 0.0227
Stage 1 = 91, CDLL = -16730.0, AR(0.77) = [0.46], Max-Change = 0.0221
Stage 1 = 92, CDLL = -16779.4, AR(0.77) = [0.47], Max-Change = 0.0369
Stage 1 = 93, CDLL = -16746.5, AR(0.77) = [0.46], Max-Change = 0.0225
Stage 1 = 94, CDLL = -16714.7, AR(0.77) = [0.48], Max-Change = 0.0297
Stage 1 = 95, CDLL = -16693.0, AR(0.77) = [0.46], Max-Change = 0.0202
Stage 1 = 96, CDLL = -16657.4, AR(0.77) = [0.46], Max-Change = 0.0246
Stage 1 = 97, CDLL = -16670.0, AR(0.77) = [0.45], Max-Change = 0.0154
Stage 1 = 98, CDLL = -16645.7, AR(0.77) = [0.46], Max-Change = 0.0311
Stage 1 = 99, CDLL = -16729.0, AR(0.77) = [0.47], Max-Change = 0.0169
Stage 1 = 100, CDLL = -16674.7, AR(0.77) = [0.46], Max-Change = 0.0121
Stage 1 = 101, CDLL = -16733.5, AR(0.77) = [0.48], Max-Change = 0.0178
Stage 1 = 102, CDLL = -16608.7, AR(0.77) = [0.46], Max-Change = 0.0218
Stage 1 = 103, CDLL = -16700.6, AR(0.77) = [0.46], Max-Change = 0.0219
Stage 1 = 104, CDLL = -16661.6, AR(0.77) = [0.46], Max-Change = 0.0172
Stage 1 = 105, CDLL = -16651.6, AR(0.77) = [0.46], Max-Change = 0.0222
Stage 1 = 106, CDLL = -16579.3, AR(0.77) = [0.45], Max-Change = 0.0261
Stage 1 = 107, CDLL = -16607.3, AR(0.77) = [0.45], Max-Change = 0.0327
Stage 1 = 108, CDLL = -16608.0, AR(0.77) = [0.48], Max-Change = 0.0243
Stage 1 = 109, CDLL = -16691.1, AR(0.77) = [0.46], Max-Change = 0.0299
Stage 1 = 110, CDLL = -16705.8, AR(0.77) = [0.45], Max-Change = 0.0117
Stage 1 = 111, CDLL = -16617.1, AR(0.77) = [0.46], Max-Change = 0.0210
Stage 1 = 112, CDLL = -16606.6, AR(0.77) = [0.47], Max-Change = 0.0216
Stage 1 = 113, CDLL = -16682.8, AR(0.77) = [0.45], Max-Change = 0.0148
Stage 1 = 114, CDLL = -16650.7, AR(0.77) = [0.46], Max-Change = 0.0162
Stage 1 = 115, CDLL = -16713.5, AR(0.77) = [0.46], Max-Change = 0.0118
Stage 1 = 116, CDLL = -16756.0, AR(0.77) = [0.46], Max-Change = 0.0257
Stage 1 = 117, CDLL = -16604.4, AR(0.77) = [0.48], Max-Change = 0.0192
Stage 1 = 118, CDLL = -16649.5, AR(0.77) = [0.45], Max-Change = 0.0155
Stage 1 = 119, CDLL = -16652.9, AR(0.77) = [0.47], Max-Change = 0.0144
Stage 1 = 120, CDLL = -16624.7, AR(0.77) = [0.48], Max-Change = 0.0139
Stage 1 = 121, CDLL = -16673.0, AR(0.77) = [0.46], Max-Change = 0.0282
Stage 1 = 122, CDLL = -16702.9, AR(0.77) = [0.48], Max-Change = 0.0157
Stage 1 = 123, CDLL = -16691.9, AR(0.77) = [0.48], Max-Change = 0.0119
Stage 1 = 124, CDLL = -16665.6, AR(0.77) = [0.47], Max-Change = 0.0175
Stage 1 = 125, CDLL = -16666.5, AR(0.77) = [0.46], Max-Change = 0.0222
Stage 1 = 126, CDLL = -16622.9, AR(0.77) = [0.45], Max-Change = 0.0141
Stage 1 = 127, CDLL = -16689.8, AR(0.77) = [0.45], Max-Change = 0.0154
Stage 1 = 128, CDLL = -16633.6, AR(0.77) = [0.46], Max-Change = 0.0162
Stage 1 = 129, CDLL = -16738.1, AR(0.77) = [0.46], Max-Change = 0.0216
Stage 1 = 130, CDLL = -16713.5, AR(0.77) = [0.45], Max-Change = 0.0158
Stage 1 = 131, CDLL = -16740.1, AR(0.77) = [0.48], Max-Change = 0.0186
Stage 1 = 132, CDLL = -16805.4, AR(0.77) = [0.48], Max-Change = 0.0416
Stage 1 = 133, CDLL = -16732.9, AR(0.77) = [0.47], Max-Change = 0.0095
Stage 1 = 134, CDLL = -16729.6, AR(0.77) = [0.47], Max-Change = 0.0196
Stage 1 = 135, CDLL = -16741.1, AR(0.77) = [0.46], Max-Change = 0.0250
Stage 1 = 136, CDLL = -16691.5, AR(0.77) = [0.48], Max-Change = 0.0201
Stage 1 = 137, CDLL = -16669.7, AR(0.77) = [0.46], Max-Change = 0.0179
Stage 1 = 138, CDLL = -16650.0, AR(0.77) = [0.44], Max-Change = 0.0382
Stage 1 = 139, CDLL = -16693.1, AR(0.77) = [0.48], Max-Change = 0.0178
Stage 1 = 140, CDLL = -16692.6, AR(0.77) = [0.48], Max-Change = 0.0179
Stage 1 = 141, CDLL = -16689.1, AR(0.77) = [0.46], Max-Change = 0.0406
Stage 1 = 142, CDLL = -16696.7, AR(0.77) = [0.47], Max-Change = 0.0150
Stage 1 = 143, CDLL = -16730.1, AR(0.77) = [0.48], Max-Change = 0.0133
Stage 1 = 144, CDLL = -16709.3, AR(0.77) = [0.43], Max-Change = 0.0245
Stage 1 = 145, CDLL = -16641.9, AR(0.77) = [0.47], Max-Change = 0.0260
Stage 1 = 146, CDLL = -16666.8, AR(0.77) = [0.46], Max-Change = 0.0205
Stage 1 = 147, CDLL = -16576.6, AR(0.77) = [0.45], Max-Change = 0.0333
Stage 1 = 148, CDLL = -16574.3, AR(0.77) = [0.47], Max-Change = 0.0203
Stage 1 = 149, CDLL = -16689.1, AR(0.77) = [0.47], Max-Change = 0.0321
Stage 1 = 150, CDLL = -16723.1, AR(1.14) = [0.38], Max-Change = 0.0194
Stage 2 = 1, CDLL = -16694.5, AR(1.14) = [0.40], Max-Change = 0.0162
Stage 2 = 2, CDLL = -16689.1, AR(1.14) = [0.38], Max-Change = 0.0227
Stage 2 = 3, CDLL = -16678.0, AR(1.14) = [0.39], Max-Change = 0.0255
Stage 2 = 4, CDLL = -16655.6, AR(1.14) = [0.38], Max-Change = 0.0238
Stage 2 = 5, CDLL = -16637.0, AR(1.14) = [0.39], Max-Change = 0.0154
Stage 2 = 6, CDLL = -16627.7, AR(1.14) = [0.41], Max-Change = 0.0277
Stage 2 = 7, CDLL = -16681.2, AR(1.14) = [0.41], Max-Change = 0.0338
Stage 2 = 8, CDLL = -16603.7, AR(1.14) = [0.38], Max-Change = 0.0178
Stage 2 = 9, CDLL = -16636.2, AR(1.14) = [0.39], Max-Change = 0.0137
Stage 2 = 10, CDLL = -16618.3, AR(1.14) = [0.38], Max-Change = 0.0175
Stage 2 = 11, CDLL = -16666.8, AR(1.14) = [0.38], Max-Change = 0.0306
Stage 2 = 12, CDLL = -16692.7, AR(1.14) = [0.38], Max-Change = 0.0112
Stage 2 = 13, CDLL = -16660.5, AR(1.14) = [0.39], Max-Change = 0.0157
Stage 2 = 14, CDLL = -16662.4, AR(1.14) = [0.39], Max-Change = 0.0224
Stage 2 = 15, CDLL = -16665.9, AR(1.14) = [0.40], Max-Change = 0.0134
Stage 2 = 16, CDLL = -16755.4, AR(1.14) = [0.39], Max-Change = 0.0180
Stage 2 = 17, CDLL = -16617.0, AR(1.14) = [0.39], Max-Change = 0.0337
Stage 2 = 18, CDLL = -16661.5, AR(1.14) = [0.39], Max-Change = 0.0228
Stage 2 = 19, CDLL = -16735.4, AR(1.14) = [0.39], Max-Change = 0.0167
Stage 2 = 20, CDLL = -16684.7, AR(1.14) = [0.38], Max-Change = 0.0171
Stage 2 = 21, CDLL = -16713.6, AR(1.14) = [0.41], Max-Change = 0.0219
Stage 2 = 22, CDLL = -16707.6, AR(1.14) = [0.39], Max-Change = 0.0234
Stage 2 = 23, CDLL = -16734.7, AR(1.14) = [0.39], Max-Change = 0.0183
Stage 2 = 24, CDLL = -16693.7, AR(1.14) = [0.38], Max-Change = 0.0198
Stage 2 = 25, CDLL = -16735.4, AR(1.14) = [0.39], Max-Change = 0.0145
Stage 2 = 26, CDLL = -16658.8, AR(1.14) = [0.40], Max-Change = 0.0128
Stage 2 = 27, CDLL = -16690.0, AR(1.14) = [0.40], Max-Change = 0.0124
Stage 2 = 28, CDLL = -16687.1, AR(1.14) = [0.40], Max-Change = 0.0139
Stage 2 = 29, CDLL = -16661.1, AR(1.14) = [0.39], Max-Change = 0.0169
Stage 2 = 30, CDLL = -16625.5, AR(1.14) = [0.39], Max-Change = 0.0131
Stage 2 = 31, CDLL = -16678.7, AR(1.14) = [0.40], Max-Change = 0.0200
Stage 2 = 32, CDLL = -16649.5, AR(1.14) = [0.38], Max-Change = 0.0172
Stage 2 = 33, CDLL = -16671.0, AR(1.14) = [0.37], Max-Change = 0.0194
Stage 2 = 34, CDLL = -16651.4, AR(1.14) = [0.37], Max-Change = 0.0125
Stage 2 = 35, CDLL = -16701.8, AR(1.14) = [0.40], Max-Change = 0.0109
Stage 2 = 36, CDLL = -16651.0, AR(1.14) = [0.42], Max-Change = 0.0216
Stage 2 = 37, CDLL = -16620.6, AR(1.14) = [0.38], Max-Change = 0.0184
Stage 2 = 38, CDLL = -16661.5, AR(1.14) = [0.41], Max-Change = 0.0217
Stage 2 = 39, CDLL = -16653.6, AR(1.14) = [0.39], Max-Change = 0.0198
Stage 2 = 40, CDLL = -16620.5, AR(1.14) = [0.39], Max-Change = 0.0152
Stage 2 = 41, CDLL = -16592.1, AR(1.14) = [0.37], Max-Change = 0.0240
Stage 2 = 42, CDLL = -16599.9, AR(1.14) = [0.38], Max-Change = 0.0312
Stage 2 = 43, CDLL = -16620.1, AR(1.14) = [0.37], Max-Change = 0.0222
Stage 2 = 44, CDLL = -16646.9, AR(1.14) = [0.41], Max-Change = 0.0144
Stage 2 = 45, CDLL = -16566.3, AR(1.14) = [0.38], Max-Change = 0.0211
Stage 2 = 46, CDLL = -16635.8, AR(1.14) = [0.38], Max-Change = 0.0184
Stage 2 = 47, CDLL = -16555.6, AR(1.14) = [0.38], Max-Change = 0.0148
Stage 2 = 48, CDLL = -16642.9, AR(1.14) = [0.38], Max-Change = 0.0275
Stage 2 = 49, CDLL = -16573.2, AR(1.14) = [0.39], Max-Change = 0.0288
Stage 2 = 50, CDLL = -16570.1, AR(1.14) = [0.38], Max-Change = 0.0243
Stage 2 = 51, CDLL = -16618.2, AR(1.14) = [0.39], Max-Change = 0.0297
Stage 2 = 52, CDLL = -16637.5, AR(1.14) = [0.40], Max-Change = 0.0182
Stage 2 = 53, CDLL = -16567.0, AR(1.14) = [0.39], Max-Change = 0.0234
Stage 2 = 54, CDLL = -16630.7, AR(1.14) = [0.37], Max-Change = 0.0222
Stage 2 = 55, CDLL = -16620.3, AR(1.14) = [0.38], Max-Change = 0.0224
Stage 2 = 56, CDLL = -16589.7, AR(1.14) = [0.39], Max-Change = 0.0108
Stage 2 = 57, CDLL = -16579.0, AR(1.14) = [0.37], Max-Change = 0.0256
Stage 2 = 58, CDLL = -16566.1, AR(1.14) = [0.40], Max-Change = 0.0220
Stage 2 = 59, CDLL = -16545.3, AR(1.14) = [0.37], Max-Change = 0.0121
Stage 2 = 60, CDLL = -16614.6, AR(1.14) = [0.39], Max-Change = 0.0171
Stage 2 = 61, CDLL = -16627.5, AR(1.14) = [0.38], Max-Change = 0.0185
Stage 2 = 62, CDLL = -16692.3, AR(1.14) = [0.35], Max-Change = 0.0222
Stage 2 = 63, CDLL = -16662.8, AR(1.14) = [0.39], Max-Change = 0.0169
Stage 2 = 64, CDLL = -16681.7, AR(1.14) = [0.39], Max-Change = 0.0185
Stage 2 = 65, CDLL = -16649.5, AR(1.14) = [0.38], Max-Change = 0.0151
Stage 2 = 66, CDLL = -16584.1, AR(1.14) = [0.40], Max-Change = 0.0265
Stage 2 = 67, CDLL = -16638.0, AR(1.14) = [0.39], Max-Change = 0.0176
Stage 2 = 68, CDLL = -16661.4, AR(1.14) = [0.40], Max-Change = 0.0196
Stage 2 = 69, CDLL = -16706.5, AR(1.14) = [0.40], Max-Change = 0.0352
Stage 2 = 70, CDLL = -16703.1, AR(1.14) = [0.40], Max-Change = 0.0351
Stage 2 = 71, CDLL = -16695.4, AR(1.14) = [0.38], Max-Change = 0.0277
Stage 2 = 72, CDLL = -16729.4, AR(1.14) = [0.40], Max-Change = 0.0267
Stage 2 = 73, CDLL = -16591.6, AR(1.14) = [0.39], Max-Change = 0.0295
Stage 2 = 74, CDLL = -16688.9, AR(1.14) = [0.38], Max-Change = 0.0144
Stage 2 = 75, CDLL = -16659.3, AR(1.14) = [0.39], Max-Change = 0.0158
Stage 2 = 76, CDLL = -16716.4, AR(1.14) = [0.39], Max-Change = 0.0126
Stage 2 = 77, CDLL = -16618.8, AR(1.14) = [0.37], Max-Change = 0.0289
Stage 2 = 78, CDLL = -16595.8, AR(1.14) = [0.39], Max-Change = 0.0357
Stage 2 = 79, CDLL = -16611.1, AR(1.14) = [0.37], Max-Change = 0.0160
Stage 2 = 80, CDLL = -16644.5, AR(1.14) = [0.41], Max-Change = 0.0232
Stage 2 = 81, CDLL = -16642.2, AR(1.14) = [0.38], Max-Change = 0.0216
Stage 2 = 82, CDLL = -16627.6, AR(1.14) = [0.40], Max-Change = 0.0305
Stage 2 = 83, CDLL = -16644.7, AR(1.14) = [0.38], Max-Change = 0.0152
Stage 2 = 84, CDLL = -16704.5, AR(1.14) = [0.38], Max-Change = 0.0200
Stage 2 = 85, CDLL = -16653.3, AR(1.14) = [0.38], Max-Change = 0.0141
Stage 2 = 86, CDLL = -16701.6, AR(1.14) = [0.37], Max-Change = 0.0248
Stage 2 = 87, CDLL = -16658.4, AR(1.14) = [0.39], Max-Change = 0.0113
Stage 2 = 88, CDLL = -16723.3, AR(1.14) = [0.39], Max-Change = 0.0354
Stage 2 = 89, CDLL = -16795.6, AR(1.14) = [0.39], Max-Change = 0.0255
Stage 2 = 90, CDLL = -16696.7, AR(1.14) = [0.39], Max-Change = 0.0222
Stage 2 = 91, CDLL = -16620.5, AR(1.14) = [0.38], Max-Change = 0.0200
Stage 2 = 92, CDLL = -16675.2, AR(1.14) = [0.40], Max-Change = 0.0290
Stage 2 = 93, CDLL = -16691.1, AR(1.14) = [0.40], Max-Change = 0.0151
Stage 2 = 94, CDLL = -16664.1, AR(1.14) = [0.40], Max-Change = 0.0204
Stage 2 = 95, CDLL = -16586.8, AR(1.14) = [0.40], Max-Change = 0.0165
Stage 2 = 96, CDLL = -16635.4, AR(1.14) = [0.39], Max-Change = 0.0108
Stage 2 = 97, CDLL = -16696.3, AR(1.14) = [0.39], Max-Change = 0.0150
Stage 2 = 98, CDLL = -16684.1, AR(1.14) = [0.40], Max-Change = 0.0179
Stage 2 = 99, CDLL = -16570.3, AR(1.14) = [0.39], Max-Change = 0.0199
Stage 2 = 100, CDLL = -16552.7, AR(1.14) = [0.38], Max-Change = 0.0162
Stage 3 = 1, CDLL = -16577.1, AR(1.14) = [0.39], gam = 0.0000, Max-Change = 0.0000
Stage 3 = 2, CDLL = -16694.9, AR(1.14) = [0.40], gam = 0.1778, Max-Change = 0.0223
Stage 3 = 3, CDLL = -16570.1, AR(1.14) = [0.40], gam = 0.1057, Max-Change = 0.0065
Stage 3 = 4, CDLL = -16671.9, AR(1.14) = [0.39], gam = 0.0780, Max-Change = 0.0059
Stage 3 = 5, CDLL = -16670.8, AR(1.14) = [0.40], gam = 0.0629, Max-Change = 0.0047
Stage 3 = 6, CDLL = -16719.7, AR(1.14) = [0.38], gam = 0.0532, Max-Change = 0.0060
Stage 3 = 7, CDLL = -16680.7, AR(1.14) = [0.40], gam = 0.0464, Max-Change = 0.0031
Stage 3 = 8, CDLL = -16633.5, AR(1.14) = [0.39], gam = 0.0413, Max-Change = 0.0042
Stage 3 = 9, CDLL = -16596.0, AR(1.14) = [0.38], gam = 0.0374, Max-Change = 0.0036
Stage 3 = 10, CDLL = -16660.2, AR(1.14) = [0.40], gam = 0.0342, Max-Change = 0.0034
Stage 3 = 11, CDLL = -16739.0, AR(1.14) = [0.39], gam = 0.0316, Max-Change = 0.0055
Stage 3 = 12, CDLL = -16651.8, AR(1.14) = [0.37], gam = 0.0294, Max-Change = 0.0032
Stage 3 = 13, CDLL = -16590.8, AR(1.14) = [0.38], gam = 0.0276, Max-Change = 0.0038
Stage 3 = 14, CDLL = -16685.0, AR(1.14) = [0.37], gam = 0.0260, Max-Change = 0.0030
Stage 3 = 15, CDLL = -16654.6, AR(1.14) = [0.37], gam = 0.0246, Max-Change = 0.0022
Stage 3 = 16, CDLL = -16705.6, AR(1.14) = [0.38], gam = 0.0233, Max-Change = 0.0021
Stage 3 = 17, CDLL = -16622.3, AR(1.14) = [0.39], gam = 0.0222, Max-Change = 0.0018
Stage 3 = 18, CDLL = -16656.3, AR(1.14) = [0.39], gam = 0.0212, Max-Change = 0.0031
Stage 3 = 19, CDLL = -16630.2, AR(1.14) = [0.39], gam = 0.0203, Max-Change = 0.0018
Stage 3 = 20, CDLL = -16644.8, AR(1.14) = [0.40], gam = 0.0195, Max-Change = 0.0012
Stage 3 = 21, CDLL = -16561.6, AR(1.14) = [0.39], gam = 0.0188, Max-Change = 0.0012
Stage 3 = 22, CDLL = -16580.3, AR(1.14) = [0.37], gam = 0.0181, Max-Change = 0.0017
Stage 3 = 23, CDLL = -16582.6, AR(1.14) = [0.39], gam = 0.0175, Max-Change = 0.0013
Stage 3 = 24, CDLL = -16702.5, AR(1.14) = [0.40], gam = 0.0169, Max-Change = 0.0008
Stage 3 = 25, CDLL = -16713.8, AR(1.14) = [0.38], gam = 0.0164, Max-Change = 0.0019
Stage 3 = 26, CDLL = -16668.4, AR(1.14) = [0.39], gam = 0.0159, Max-Change = 0.0010
Stage 3 = 27, CDLL = -16599.2, AR(1.14) = [0.38], gam = 0.0154, Max-Change = 0.0018
Stage 3 = 28, CDLL = -16593.2, AR(1.14) = [0.41], gam = 0.0150, Max-Change = 0.0018
Stage 3 = 29, CDLL = -16595.8, AR(1.14) = [0.39], gam = 0.0146, Max-Change = 0.0020
Stage 3 = 30, CDLL = -16559.0, AR(1.14) = [0.39], gam = 0.0142, Max-Change = 0.0018
Stage 3 = 31, CDLL = -16606.1, AR(1.14) = [0.38], gam = 0.0139, Max-Change = 0.0018
Stage 3 = 32, CDLL = -16681.8, AR(1.14) = [0.40], gam = 0.0135, Max-Change = 0.0024
Stage 3 = 33, CDLL = -16700.2, AR(1.14) = [0.40], gam = 0.0132, Max-Change = 0.0018
Stage 3 = 34, CDLL = -16625.9, AR(1.14) = [0.39], gam = 0.0129, Max-Change = 0.0009
Stage 3 = 35, CDLL = -16704.5, AR(1.14) = [0.38], gam = 0.0126, Max-Change = 0.0008
Stage 3 = 36, CDLL = -16642.3, AR(1.14) = [0.37], gam = 0.0124, Max-Change = 0.0015
Stage 3 = 37, CDLL = -16618.9, AR(1.14) = [0.39], gam = 0.0121, Max-Change = 0.0013
Stage 3 = 38, CDLL = -16711.5, AR(1.14) = [0.42], gam = 0.0119, Max-Change = 0.0015
Stage 3 = 39, CDLL = -16651.9, AR(1.14) = [0.38], gam = 0.0116, Max-Change = 0.0011
Stage 3 = 40, CDLL = -16628.1, AR(1.14) = [0.38], gam = 0.0114, Max-Change = 0.0010
Stage 3 = 41, CDLL = -16669.0, AR(1.14) = [0.38], gam = 0.0112, Max-Change = 0.0027
Stage 3 = 42, CDLL = -16665.7, AR(1.14) = [0.38], gam = 0.0110, Max-Change = 0.0009
Stage 3 = 43, CDLL = -16667.2, AR(1.14) = [0.40], gam = 0.0108, Max-Change = 0.0012
Stage 3 = 44, CDLL = -16590.8, AR(1.14) = [0.39], gam = 0.0106, Max-Change = 0.0013
Stage 3 = 45, CDLL = -16604.2, AR(1.14) = [0.39], gam = 0.0104, Max-Change = 0.0010
Stage 3 = 46, CDLL = -16733.1, AR(1.14) = [0.38], gam = 0.0102, Max-Change = 0.0015
Stage 3 = 47, CDLL = -16655.8, AR(1.14) = [0.38], gam = 0.0101, Max-Change = 0.0009
Stage 3 = 48, CDLL = -16702.2, AR(1.14) = [0.39], gam = 0.0099, Max-Change = 0.0007
Stage 3 = 49, CDLL = -16670.9, AR(1.14) = [0.41], gam = 0.0098, Max-Change = 0.0013
Stage 3 = 50, CDLL = -16650.5, AR(1.14) = [0.38], gam = 0.0096, Max-Change = 0.0020
Stage 3 = 51, CDLL = -16715.9, AR(1.14) = [0.38], gam = 0.0095, Max-Change = 0.0016
Stage 3 = 52, CDLL = -16658.5, AR(1.14) = [0.38], gam = 0.0093, Max-Change = 0.0012
Stage 3 = 53, CDLL = -16681.3, AR(1.14) = [0.36], gam = 0.0092, Max-Change = 0.0008
Stage 3 = 54, CDLL = -16600.1, AR(1.14) = [0.39], gam = 0.0091, Max-Change = 0.0005
Stage 3 = 55, CDLL = -16611.7, AR(1.14) = [0.38], gam = 0.0089, Max-Change = 0.0008

#> 
#> Calculating log-likelihood...
coef(mod1)
#> $Item_1
#>        a1 a2      d g u
#> par 1.798  0 -0.991 0 1
#> 
#> $Item_2
#>        a1 a2      d g u
#> par 0.488  0 -1.527 0 1
#> 
#> $Item_3
#>      a1 a2     d g u
#> par 0.9  0 1.519 0 1
#> 
#> $Item_4
#>        a1    a2     d g u
#> par 1.042 0.557 0.088 0 1
#> 
#> $Item_5
#>     a1    a2    d1   d2     d3
#> par  0 1.462 2.856 1.91 -0.523
#> 
#> $Item_6
#>     a1    a2    d1    d2     d3
#> par  0 0.635 2.494 1.007 -1.047
#> 
#> $Item_7
#>     a1    a2 d1     d2
#> par  0 1.018  2 -0.025
#> 
#> $Item_8
#>     a1    a2     d g u
#> par  0 0.959 0.957 0 1
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 COV_11 COV_21 COV_22
#> par      0      0      1  0.318      1
#> 
summary(mod1)
#>           F1    F2    h2
#> Item_1 0.726       0.527
#> Item_2 0.276       0.076
#> Item_3 0.468       0.219
#> Item_4 0.503 0.269 0.325
#> Item_5       0.651 0.424
#> Item_6       0.350 0.122
#> Item_7       0.513 0.264
#> Item_8       0.491 0.241
#> 
#> SS loadings:  1.075 1.124 
#> Proportion Var:  0.134 0.14 
#> 
#> Factor correlations: 
#> 
#>       F1 F2
#> F1 1.000   
#> F2 0.318  1
residuals(mod1)
#> LD matrix (lower triangle) and standardized residual correlations (upper triangle)
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.040  -0.018   0.004   0.005   0.034   0.059 
#> 
#>        Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8
#> Item_1        -0.002 -0.009  0.003  0.006  0.059 -0.007 -0.028
#> Item_2  0.006         0.014 -0.017  0.046  0.050  0.019  0.010
#> Item_3  0.158  0.412         0.006  0.034  0.027  0.034 -0.019
#> Item_4  0.020  0.609  0.068        -0.035  0.040  0.045 -0.039
#> Item_5  0.064  4.239  2.274  2.466        -0.032 -0.040  0.038
#> Item_6  7.035  5.086  1.415  3.234  6.022        -0.034 -0.016
#> Item_7  0.105  0.746  2.307  4.093  6.393  4.612        -0.014
#> Item_8  1.531  0.220  0.689  3.029  2.913  0.488  0.400       

#####
# bifactor
model.3 <- '
  G = 1-8
  F1 = 1-4
  F2 = 5-8'

mod3 <- mirt(dataset,model.3, method = 'MHRM')
#> 
Stage 1 = 1, CDLL = -19647.0, AR(1.00) = [0.36], Max-Change = 0.1193
Stage 1 = 2, CDLL = -19640.5, AR(1.00) = [0.34], Max-Change = 0.0933
Stage 1 = 3, CDLL = -19637.9, AR(1.00) = [0.36], Max-Change = 0.0742
Stage 1 = 4, CDLL = -19635.4, AR(1.00) = [0.34], Max-Change = 0.0508
Stage 1 = 5, CDLL = -19703.3, AR(1.00) = [0.34], Max-Change = 0.0523
Stage 1 = 6, CDLL = -19640.5, AR(1.00) = [0.34], Max-Change = 0.0528
Stage 1 = 7, CDLL = -19688.1, AR(1.00) = [0.36], Max-Change = 0.0400
Stage 1 = 8, CDLL = -19627.4, AR(1.00) = [0.35], Max-Change = 0.0417
Stage 1 = 9, CDLL = -19668.7, AR(1.00) = [0.35], Max-Change = 0.0340
Stage 1 = 10, CDLL = -19717.8, AR(1.00) = [0.38], Max-Change = 0.0264
Stage 1 = 11, CDLL = -19676.2, AR(1.00) = [0.36], Max-Change = 0.0213
Stage 1 = 12, CDLL = -19636.0, AR(1.00) = [0.36], Max-Change = 0.0350
Stage 1 = 13, CDLL = -19605.4, AR(1.00) = [0.36], Max-Change = 0.0216
Stage 1 = 14, CDLL = -19649.6, AR(1.00) = [0.36], Max-Change = 0.0240
Stage 1 = 15, CDLL = -19654.7, AR(1.00) = [0.36], Max-Change = 0.0270
Stage 1 = 16, CDLL = -19648.3, AR(1.00) = [0.35], Max-Change = 0.0220
Stage 1 = 17, CDLL = -19667.7, AR(1.00) = [0.36], Max-Change = 0.0262
Stage 1 = 18, CDLL = -19818.9, AR(1.00) = [0.34], Max-Change = 0.0247
Stage 1 = 19, CDLL = -19888.8, AR(1.00) = [0.37], Max-Change = 0.0275
Stage 1 = 20, CDLL = -19726.6, AR(1.00) = [0.38], Max-Change = 0.0261
Stage 1 = 21, CDLL = -19711.4, AR(1.00) = [0.37], Max-Change = 0.0194
Stage 1 = 22, CDLL = -19654.5, AR(1.00) = [0.36], Max-Change = 0.0284
Stage 1 = 23, CDLL = -19643.0, AR(1.00) = [0.35], Max-Change = 0.0234
Stage 1 = 24, CDLL = -19639.5, AR(1.00) = [0.35], Max-Change = 0.0139
Stage 1 = 25, CDLL = -19641.0, AR(1.00) = [0.37], Max-Change = 0.0312
Stage 1 = 26, CDLL = -19589.7, AR(1.00) = [0.36], Max-Change = 0.0204
Stage 1 = 27, CDLL = -19614.5, AR(1.00) = [0.35], Max-Change = 0.0295
Stage 1 = 28, CDLL = -19666.3, AR(1.00) = [0.37], Max-Change = 0.0169
Stage 1 = 29, CDLL = -19601.1, AR(1.00) = [0.38], Max-Change = 0.0214
Stage 1 = 30, CDLL = -19604.8, AR(1.00) = [0.38], Max-Change = 0.0225
Stage 1 = 31, CDLL = -19605.4, AR(1.00) = [0.36], Max-Change = 0.0168
Stage 1 = 32, CDLL = -19705.1, AR(1.00) = [0.35], Max-Change = 0.0211
Stage 1 = 33, CDLL = -19636.5, AR(1.00) = [0.34], Max-Change = 0.0192
Stage 1 = 34, CDLL = -19671.1, AR(1.00) = [0.35], Max-Change = 0.0272
Stage 1 = 35, CDLL = -19700.2, AR(1.00) = [0.37], Max-Change = 0.0184
Stage 1 = 36, CDLL = -19674.6, AR(1.00) = [0.35], Max-Change = 0.0189
Stage 1 = 37, CDLL = -19733.9, AR(1.00) = [0.36], Max-Change = 0.0165
Stage 1 = 38, CDLL = -19664.2, AR(1.00) = [0.37], Max-Change = 0.0149
Stage 1 = 39, CDLL = -19653.8, AR(1.00) = [0.35], Max-Change = 0.0211
Stage 1 = 40, CDLL = -19564.3, AR(1.00) = [0.35], Max-Change = 0.0257
Stage 1 = 41, CDLL = -19579.7, AR(1.00) = [0.35], Max-Change = 0.0276
Stage 1 = 42, CDLL = -19661.2, AR(1.00) = [0.34], Max-Change = 0.0185
Stage 1 = 43, CDLL = -19678.7, AR(1.00) = [0.36], Max-Change = 0.0222
Stage 1 = 44, CDLL = -19636.7, AR(1.00) = [0.35], Max-Change = 0.0424
Stage 1 = 45, CDLL = -19633.4, AR(1.00) = [0.33], Max-Change = 0.0164
Stage 1 = 46, CDLL = -19658.6, AR(1.00) = [0.35], Max-Change = 0.0202
Stage 1 = 47, CDLL = -19639.1, AR(1.00) = [0.35], Max-Change = 0.0165
Stage 1 = 48, CDLL = -19653.2, AR(1.00) = [0.36], Max-Change = 0.0283
Stage 1 = 49, CDLL = -19683.7, AR(1.00) = [0.36], Max-Change = 0.0166
Stage 1 = 50, CDLL = -19664.5, AR(1.00) = [0.37], Max-Change = 0.0172
Stage 1 = 51, CDLL = -19600.6, AR(1.00) = [0.33], Max-Change = 0.0147
Stage 1 = 52, CDLL = -19551.1, AR(1.00) = [0.36], Max-Change = 0.0172
Stage 1 = 53, CDLL = -19630.4, AR(1.00) = [0.35], Max-Change = 0.0247
Stage 1 = 54, CDLL = -19599.9, AR(1.00) = [0.36], Max-Change = 0.0206
Stage 1 = 55, CDLL = -19592.4, AR(1.00) = [0.35], Max-Change = 0.0188
Stage 1 = 56, CDLL = -19579.6, AR(1.00) = [0.35], Max-Change = 0.0143
Stage 1 = 57, CDLL = -19592.4, AR(1.00) = [0.34], Max-Change = 0.0206
Stage 1 = 58, CDLL = -19558.9, AR(1.00) = [0.36], Max-Change = 0.0345
Stage 1 = 59, CDLL = -19611.9, AR(1.00) = [0.36], Max-Change = 0.0170
Stage 1 = 60, CDLL = -19582.4, AR(1.00) = [0.33], Max-Change = 0.0178
Stage 1 = 61, CDLL = -19640.5, AR(1.00) = [0.36], Max-Change = 0.0207
Stage 1 = 62, CDLL = -19580.8, AR(1.00) = [0.36], Max-Change = 0.0140
Stage 1 = 63, CDLL = -19565.4, AR(1.00) = [0.34], Max-Change = 0.0228
Stage 1 = 64, CDLL = -19576.9, AR(1.00) = [0.34], Max-Change = 0.0221
Stage 1 = 65, CDLL = -19572.2, AR(1.00) = [0.35], Max-Change = 0.0208
Stage 1 = 66, CDLL = -19552.8, AR(1.00) = [0.35], Max-Change = 0.0199
Stage 1 = 67, CDLL = -19413.6, AR(1.00) = [0.35], Max-Change = 0.0193
Stage 1 = 68, CDLL = -19452.0, AR(1.00) = [0.36], Max-Change = 0.0195
Stage 1 = 69, CDLL = -19530.8, AR(1.00) = [0.36], Max-Change = 0.0242
Stage 1 = 70, CDLL = -19584.2, AR(1.00) = [0.34], Max-Change = 0.0224
Stage 1 = 71, CDLL = -19543.8, AR(1.00) = [0.33], Max-Change = 0.0146
Stage 1 = 72, CDLL = -19546.7, AR(1.00) = [0.34], Max-Change = 0.0218
Stage 1 = 73, CDLL = -19495.1, AR(1.00) = [0.35], Max-Change = 0.0299
Stage 1 = 74, CDLL = -19526.3, AR(1.00) = [0.35], Max-Change = 0.0335
Stage 1 = 75, CDLL = -19573.4, AR(1.00) = [0.37], Max-Change = 0.0223
Stage 1 = 76, CDLL = -19531.1, AR(1.00) = [0.34], Max-Change = 0.0232
Stage 1 = 77, CDLL = -19514.1, AR(1.00) = [0.35], Max-Change = 0.0204
Stage 1 = 78, CDLL = -19642.9, AR(1.00) = [0.37], Max-Change = 0.0239
Stage 1 = 79, CDLL = -19516.3, AR(1.00) = [0.35], Max-Change = 0.0184
Stage 1 = 80, CDLL = -19539.4, AR(1.00) = [0.35], Max-Change = 0.0242
Stage 1 = 81, CDLL = -19597.6, AR(1.00) = [0.34], Max-Change = 0.0129
Stage 1 = 82, CDLL = -19566.1, AR(1.00) = [0.33], Max-Change = 0.0252
Stage 1 = 83, CDLL = -19524.4, AR(1.00) = [0.34], Max-Change = 0.0173
Stage 1 = 84, CDLL = -19517.0, AR(1.00) = [0.34], Max-Change = 0.0205
Stage 1 = 85, CDLL = -19520.9, AR(1.00) = [0.37], Max-Change = 0.0172
Stage 1 = 86, CDLL = -19560.5, AR(1.00) = [0.32], Max-Change = 0.0183
Stage 1 = 87, CDLL = -19615.9, AR(1.00) = [0.35], Max-Change = 0.0207
Stage 1 = 88, CDLL = -19654.7, AR(1.00) = [0.34], Max-Change = 0.0238
Stage 1 = 89, CDLL = -19614.2, AR(1.00) = [0.34], Max-Change = 0.0104
Stage 1 = 90, CDLL = -19588.2, AR(1.00) = [0.34], Max-Change = 0.0360
Stage 1 = 91, CDLL = -19544.5, AR(1.00) = [0.35], Max-Change = 0.0195
Stage 1 = 92, CDLL = -19496.4, AR(1.00) = [0.34], Max-Change = 0.0124
Stage 1 = 93, CDLL = -19508.2, AR(1.00) = [0.33], Max-Change = 0.0160
Stage 1 = 94, CDLL = -19500.9, AR(1.00) = [0.35], Max-Change = 0.0278
Stage 1 = 95, CDLL = -19498.7, AR(1.00) = [0.34], Max-Change = 0.0191
Stage 1 = 96, CDLL = -19521.8, AR(1.00) = [0.35], Max-Change = 0.0171
Stage 1 = 97, CDLL = -19576.5, AR(1.00) = [0.34], Max-Change = 0.0234
Stage 1 = 98, CDLL = -19586.7, AR(1.00) = [0.36], Max-Change = 0.0313
Stage 1 = 99, CDLL = -19540.4, AR(1.00) = [0.34], Max-Change = 0.0287
Stage 1 = 100, CDLL = -19496.2, AR(1.00) = [0.35], Max-Change = 0.0205
Stage 1 = 101, CDLL = -19413.5, AR(1.00) = [0.34], Max-Change = 0.0174
Stage 1 = 102, CDLL = -19475.5, AR(1.00) = [0.35], Max-Change = 0.0203
Stage 1 = 103, CDLL = -19468.6, AR(1.00) = [0.33], Max-Change = 0.0161
Stage 1 = 104, CDLL = -19533.0, AR(1.00) = [0.36], Max-Change = 0.0160
Stage 1 = 105, CDLL = -19496.5, AR(1.00) = [0.33], Max-Change = 0.0177
Stage 1 = 106, CDLL = -19500.3, AR(1.00) = [0.34], Max-Change = 0.0163
Stage 1 = 107, CDLL = -19431.8, AR(1.00) = [0.35], Max-Change = 0.0140
Stage 1 = 108, CDLL = -19413.8, AR(1.00) = [0.34], Max-Change = 0.0208
Stage 1 = 109, CDLL = -19430.8, AR(1.00) = [0.36], Max-Change = 0.0180
Stage 1 = 110, CDLL = -19439.1, AR(1.00) = [0.33], Max-Change = 0.0175
Stage 1 = 111, CDLL = -19518.8, AR(1.00) = [0.34], Max-Change = 0.0376
Stage 1 = 112, CDLL = -19503.0, AR(1.00) = [0.35], Max-Change = 0.0205
Stage 1 = 113, CDLL = -19555.8, AR(1.00) = [0.34], Max-Change = 0.0199
Stage 1 = 114, CDLL = -19495.2, AR(1.00) = [0.34], Max-Change = 0.0190
Stage 1 = 115, CDLL = -19400.2, AR(1.00) = [0.34], Max-Change = 0.0184
Stage 1 = 116, CDLL = -19392.6, AR(1.00) = [0.34], Max-Change = 0.0215
Stage 1 = 117, CDLL = -19502.6, AR(1.00) = [0.35], Max-Change = 0.0137
Stage 1 = 118, CDLL = -19453.4, AR(1.00) = [0.34], Max-Change = 0.0186
Stage 1 = 119, CDLL = -19493.4, AR(1.00) = [0.34], Max-Change = 0.0385
Stage 1 = 120, CDLL = -19549.1, AR(1.00) = [0.34], Max-Change = 0.0221
Stage 1 = 121, CDLL = -19529.7, AR(1.00) = [0.34], Max-Change = 0.0263
Stage 1 = 122, CDLL = -19532.9, AR(1.00) = [0.35], Max-Change = 0.0295
Stage 1 = 123, CDLL = -19603.6, AR(1.00) = [0.34], Max-Change = 0.0171
Stage 1 = 124, CDLL = -19541.4, AR(1.00) = [0.36], Max-Change = 0.0141
Stage 1 = 125, CDLL = -19406.2, AR(1.00) = [0.34], Max-Change = 0.0200
Stage 1 = 126, CDLL = -19451.9, AR(1.00) = [0.32], Max-Change = 0.0166
Stage 1 = 127, CDLL = -19463.4, AR(1.00) = [0.34], Max-Change = 0.0154
Stage 1 = 128, CDLL = -19497.6, AR(1.00) = [0.33], Max-Change = 0.0175
Stage 1 = 129, CDLL = -19540.1, AR(1.00) = [0.34], Max-Change = 0.0147
Stage 1 = 130, CDLL = -19530.7, AR(1.00) = [0.32], Max-Change = 0.0252
Stage 1 = 131, CDLL = -19524.9, AR(1.00) = [0.33], Max-Change = 0.0185
Stage 1 = 132, CDLL = -19532.0, AR(1.00) = [0.35], Max-Change = 0.0250
Stage 1 = 133, CDLL = -19414.9, AR(1.00) = [0.35], Max-Change = 0.0155
Stage 1 = 134, CDLL = -19437.5, AR(1.00) = [0.34], Max-Change = 0.0175
Stage 1 = 135, CDLL = -19496.8, AR(1.00) = [0.34], Max-Change = 0.0152
Stage 1 = 136, CDLL = -19518.5, AR(1.00) = [0.35], Max-Change = 0.0186
Stage 1 = 137, CDLL = -19484.3, AR(1.00) = [0.35], Max-Change = 0.0123
Stage 1 = 138, CDLL = -19554.9, AR(1.00) = [0.34], Max-Change = 0.0147
Stage 1 = 139, CDLL = -19519.1, AR(1.00) = [0.35], Max-Change = 0.0285
Stage 1 = 140, CDLL = -19518.8, AR(1.00) = [0.35], Max-Change = 0.0413
Stage 1 = 141, CDLL = -19410.6, AR(1.00) = [0.35], Max-Change = 0.0179
Stage 1 = 142, CDLL = -19398.1, AR(1.00) = [0.34], Max-Change = 0.0205
Stage 1 = 143, CDLL = -19404.4, AR(1.00) = [0.34], Max-Change = 0.0170
Stage 1 = 144, CDLL = -19400.9, AR(1.00) = [0.32], Max-Change = 0.0212
Stage 1 = 145, CDLL = -19446.2, AR(1.00) = [0.34], Max-Change = 0.0110
Stage 1 = 146, CDLL = -19423.8, AR(1.00) = [0.34], Max-Change = 0.0117
Stage 1 = 147, CDLL = -19445.3, AR(1.00) = [0.35], Max-Change = 0.0207
Stage 1 = 148, CDLL = -19403.2, AR(1.00) = [0.34], Max-Change = 0.0250
Stage 1 = 149, CDLL = -19514.0, AR(1.00) = [0.35], Max-Change = 0.0211
Stage 1 = 150, CDLL = -19525.8, AR(0.76) = [0.40], Max-Change = 0.0258
Stage 2 = 1, CDLL = -19504.1, AR(0.76) = [0.39], Max-Change = 0.0226
Stage 2 = 2, CDLL = -19493.9, AR(0.76) = [0.41], Max-Change = 0.0228
Stage 2 = 3, CDLL = -19518.3, AR(0.76) = [0.41], Max-Change = 0.0219
Stage 2 = 4, CDLL = -19415.0, AR(0.76) = [0.41], Max-Change = 0.0195
Stage 2 = 5, CDLL = -19502.8, AR(0.76) = [0.41], Max-Change = 0.0237
Stage 2 = 6, CDLL = -19575.6, AR(0.76) = [0.40], Max-Change = 0.0171
Stage 2 = 7, CDLL = -19515.5, AR(0.76) = [0.39], Max-Change = 0.0114
Stage 2 = 8, CDLL = -19450.9, AR(0.76) = [0.40], Max-Change = 0.0190
Stage 2 = 9, CDLL = -19546.2, AR(0.76) = [0.39], Max-Change = 0.0175
Stage 2 = 10, CDLL = -19534.8, AR(0.76) = [0.39], Max-Change = 0.0215
Stage 2 = 11, CDLL = -19562.8, AR(0.76) = [0.41], Max-Change = 0.0205
Stage 2 = 12, CDLL = -19415.6, AR(0.76) = [0.38], Max-Change = 0.0229
Stage 2 = 13, CDLL = -19459.1, AR(0.76) = [0.40], Max-Change = 0.0190
Stage 2 = 14, CDLL = -19395.8, AR(0.76) = [0.39], Max-Change = 0.0206
Stage 2 = 15, CDLL = -19400.8, AR(0.76) = [0.40], Max-Change = 0.0207
Stage 2 = 16, CDLL = -19530.2, AR(0.76) = [0.41], Max-Change = 0.0240
Stage 2 = 17, CDLL = -19421.0, AR(0.76) = [0.40], Max-Change = 0.0262
Stage 2 = 18, CDLL = -19455.9, AR(0.76) = [0.39], Max-Change = 0.0181
Stage 2 = 19, CDLL = -19502.3, AR(0.76) = [0.41], Max-Change = 0.0288
Stage 2 = 20, CDLL = -19473.0, AR(0.76) = [0.39], Max-Change = 0.0170
Stage 2 = 21, CDLL = -19443.1, AR(0.76) = [0.39], Max-Change = 0.0421
Stage 2 = 22, CDLL = -19479.7, AR(0.76) = [0.41], Max-Change = 0.0346
Stage 2 = 23, CDLL = -19457.9, AR(0.76) = [0.39], Max-Change = 0.0192
Stage 2 = 24, CDLL = -19550.3, AR(0.76) = [0.42], Max-Change = 0.0153
Stage 2 = 25, CDLL = -19545.4, AR(0.76) = [0.41], Max-Change = 0.0332
Stage 2 = 26, CDLL = -19474.7, AR(0.76) = [0.39], Max-Change = 0.0222
Stage 2 = 27, CDLL = -19394.5, AR(0.76) = [0.40], Max-Change = 0.0380
Stage 2 = 28, CDLL = -19554.7, AR(0.76) = [0.41], Max-Change = 0.0240
Stage 2 = 29, CDLL = -19531.2, AR(0.76) = [0.39], Max-Change = 0.0228
Stage 2 = 30, CDLL = -19386.9, AR(0.76) = [0.40], Max-Change = 0.0266
Stage 2 = 31, CDLL = -19456.6, AR(0.76) = [0.39], Max-Change = 0.0253
Stage 2 = 32, CDLL = -19487.4, AR(0.76) = [0.39], Max-Change = 0.0184
Stage 2 = 33, CDLL = -19524.8, AR(0.76) = [0.37], Max-Change = 0.0213
Stage 2 = 34, CDLL = -19609.9, AR(0.76) = [0.41], Max-Change = 0.0192
Stage 2 = 35, CDLL = -19520.4, AR(0.76) = [0.41], Max-Change = 0.0250
Stage 2 = 36, CDLL = -19564.2, AR(0.76) = [0.42], Max-Change = 0.0255
Stage 2 = 37, CDLL = -19599.6, AR(0.76) = [0.41], Max-Change = 0.0186
Stage 2 = 38, CDLL = -19556.6, AR(0.76) = [0.41], Max-Change = 0.0209
Stage 2 = 39, CDLL = -19569.2, AR(0.76) = [0.40], Max-Change = 0.0234
Stage 2 = 40, CDLL = -19505.0, AR(0.76) = [0.41], Max-Change = 0.0123
Stage 2 = 41, CDLL = -19527.4, AR(0.76) = [0.40], Max-Change = 0.0238
Stage 2 = 42, CDLL = -19554.7, AR(0.76) = [0.42], Max-Change = 0.0147
Stage 2 = 43, CDLL = -19557.3, AR(0.76) = [0.39], Max-Change = 0.0184
Stage 2 = 44, CDLL = -19485.9, AR(0.76) = [0.40], Max-Change = 0.0203
Stage 2 = 45, CDLL = -19584.3, AR(0.76) = [0.40], Max-Change = 0.0224
Stage 2 = 46, CDLL = -19587.4, AR(0.76) = [0.41], Max-Change = 0.0254
Stage 2 = 47, CDLL = -19588.7, AR(0.76) = [0.41], Max-Change = 0.0217
Stage 2 = 48, CDLL = -19583.0, AR(0.76) = [0.41], Max-Change = 0.0200
Stage 2 = 49, CDLL = -19535.6, AR(0.76) = [0.41], Max-Change = 0.0193
Stage 2 = 50, CDLL = -19474.0, AR(0.76) = [0.39], Max-Change = 0.0139
Stage 2 = 51, CDLL = -19481.2, AR(0.76) = [0.39], Max-Change = 0.0208
Stage 2 = 52, CDLL = -19556.9, AR(0.76) = [0.39], Max-Change = 0.0201
Stage 2 = 53, CDLL = -19564.6, AR(0.76) = [0.38], Max-Change = 0.0183
Stage 2 = 54, CDLL = -19560.3, AR(0.76) = [0.42], Max-Change = 0.0173
Stage 2 = 55, CDLL = -19485.6, AR(0.76) = [0.37], Max-Change = 0.0186
Stage 2 = 56, CDLL = -19624.3, AR(0.76) = [0.39], Max-Change = 0.0185
Stage 2 = 57, CDLL = -19490.2, AR(0.76) = [0.40], Max-Change = 0.0156
Stage 2 = 58, CDLL = -19526.6, AR(0.76) = [0.40], Max-Change = 0.0230
Stage 2 = 59, CDLL = -19453.6, AR(0.76) = [0.41], Max-Change = 0.0235
Stage 2 = 60, CDLL = -19556.4, AR(0.76) = [0.40], Max-Change = 0.0277
Stage 2 = 61, CDLL = -19579.3, AR(0.76) = [0.41], Max-Change = 0.0326
Stage 2 = 62, CDLL = -19573.0, AR(0.76) = [0.39], Max-Change = 0.0270
Stage 2 = 63, CDLL = -19542.5, AR(0.76) = [0.42], Max-Change = 0.0219
Stage 2 = 64, CDLL = -19579.1, AR(0.76) = [0.40], Max-Change = 0.0220
Stage 2 = 65, CDLL = -19608.4, AR(0.76) = [0.41], Max-Change = 0.0192
Stage 2 = 66, CDLL = -19608.1, AR(0.76) = [0.40], Max-Change = 0.0282
Stage 2 = 67, CDLL = -19648.1, AR(0.76) = [0.41], Max-Change = 0.0164
Stage 2 = 68, CDLL = -19626.9, AR(0.76) = [0.39], Max-Change = 0.0205
Stage 2 = 69, CDLL = -19579.8, AR(0.76) = [0.40], Max-Change = 0.0183
Stage 2 = 70, CDLL = -19609.5, AR(0.76) = [0.39], Max-Change = 0.0187
Stage 2 = 71, CDLL = -19593.8, AR(0.76) = [0.40], Max-Change = 0.0171
Stage 2 = 72, CDLL = -19482.4, AR(0.76) = [0.40], Max-Change = 0.0288
Stage 2 = 73, CDLL = -19517.2, AR(0.76) = [0.38], Max-Change = 0.0447
Stage 2 = 74, CDLL = -19505.9, AR(0.76) = [0.39], Max-Change = 0.0316
Stage 2 = 75, CDLL = -19564.2, AR(0.76) = [0.40], Max-Change = 0.0187
Stage 2 = 76, CDLL = -19542.7, AR(0.76) = [0.41], Max-Change = 0.0256
Stage 2 = 77, CDLL = -19494.8, AR(0.76) = [0.40], Max-Change = 0.0257
Stage 2 = 78, CDLL = -19530.9, AR(0.76) = [0.38], Max-Change = 0.0190
Stage 2 = 79, CDLL = -19526.0, AR(0.76) = [0.40], Max-Change = 0.0158
Stage 2 = 80, CDLL = -19520.5, AR(0.76) = [0.40], Max-Change = 0.0188
Stage 2 = 81, CDLL = -19589.5, AR(0.76) = [0.40], Max-Change = 0.0127
Stage 2 = 82, CDLL = -19482.7, AR(0.76) = [0.39], Max-Change = 0.0182
Stage 2 = 83, CDLL = -19527.9, AR(0.76) = [0.41], Max-Change = 0.0192
Stage 2 = 84, CDLL = -19555.2, AR(0.76) = [0.39], Max-Change = 0.0255
Stage 2 = 85, CDLL = -19624.7, AR(0.76) = [0.41], Max-Change = 0.0197
Stage 2 = 86, CDLL = -19556.1, AR(0.76) = [0.41], Max-Change = 0.0193
Stage 2 = 87, CDLL = -19486.7, AR(0.76) = [0.39], Max-Change = 0.0186
Stage 2 = 88, CDLL = -19507.6, AR(0.76) = [0.41], Max-Change = 0.0260
Stage 2 = 89, CDLL = -19523.2, AR(0.76) = [0.42], Max-Change = 0.0241
Stage 2 = 90, CDLL = -19587.2, AR(0.76) = [0.41], Max-Change = 0.0244
Stage 2 = 91, CDLL = -19594.8, AR(0.76) = [0.42], Max-Change = 0.0153
Stage 2 = 92, CDLL = -19541.8, AR(0.76) = [0.39], Max-Change = 0.0275
Stage 2 = 93, CDLL = -19490.8, AR(0.76) = [0.40], Max-Change = 0.0190
Stage 2 = 94, CDLL = -19537.9, AR(0.76) = [0.42], Max-Change = 0.0123
Stage 2 = 95, CDLL = -19611.7, AR(0.76) = [0.39], Max-Change = 0.0203
Stage 2 = 96, CDLL = -19494.2, AR(0.76) = [0.41], Max-Change = 0.0186
Stage 2 = 97, CDLL = -19438.0, AR(0.76) = [0.38], Max-Change = 0.0207
Stage 2 = 98, CDLL = -19438.2, AR(0.76) = [0.40], Max-Change = 0.0238
Stage 2 = 99, CDLL = -19344.9, AR(0.76) = [0.40], Max-Change = 0.0244
Stage 2 = 100, CDLL = -19381.9, AR(0.76) = [0.39], Max-Change = 0.0191
Stage 3 = 1, CDLL = -19427.9, AR(0.76) = [0.37], gam = 0.0000, Max-Change = 0.0000
Stage 3 = 2, CDLL = -19492.5, AR(0.76) = [0.39], gam = 0.1778, Max-Change = 0.0231
Stage 3 = 3, CDLL = -19492.7, AR(0.76) = [0.41], gam = 0.1057, Max-Change = 0.0086
Stage 3 = 4, CDLL = -19424.6, AR(0.76) = [0.40], gam = 0.0780, Max-Change = 0.0110
Stage 3 = 5, CDLL = -19420.1, AR(0.76) = [0.38], gam = 0.0629, Max-Change = 0.0088
Stage 3 = 6, CDLL = -19512.3, AR(0.76) = [0.39], gam = 0.0532, Max-Change = 0.0069
Stage 3 = 7, CDLL = -19456.9, AR(0.76) = [0.39], gam = 0.0464, Max-Change = 0.0056
Stage 3 = 8, CDLL = -19483.0, AR(0.76) = [0.40], gam = 0.0413, Max-Change = 0.0040
Stage 3 = 9, CDLL = -19560.0, AR(0.76) = [0.41], gam = 0.0374, Max-Change = 0.0037
Stage 3 = 10, CDLL = -19539.9, AR(0.76) = [0.39], gam = 0.0342, Max-Change = 0.0025
Stage 3 = 11, CDLL = -19548.2, AR(0.76) = [0.38], gam = 0.0316, Max-Change = 0.0044
Stage 3 = 12, CDLL = -19527.8, AR(0.76) = [0.38], gam = 0.0294, Max-Change = 0.0026
Stage 3 = 13, CDLL = -19537.3, AR(0.76) = [0.40], gam = 0.0276, Max-Change = 0.0030
Stage 3 = 14, CDLL = -19557.7, AR(0.76) = [0.40], gam = 0.0260, Max-Change = 0.0030
Stage 3 = 15, CDLL = -19555.9, AR(0.76) = [0.41], gam = 0.0246, Max-Change = 0.0029
Stage 3 = 16, CDLL = -19441.3, AR(0.76) = [0.40], gam = 0.0233, Max-Change = 0.0020
Stage 3 = 17, CDLL = -19431.3, AR(0.76) = [0.39], gam = 0.0222, Max-Change = 0.0020
Stage 3 = 18, CDLL = -19434.0, AR(0.76) = [0.40], gam = 0.0212, Max-Change = 0.0019
Stage 3 = 19, CDLL = -19478.6, AR(0.76) = [0.41], gam = 0.0203, Max-Change = 0.0028
Stage 3 = 20, CDLL = -19519.7, AR(0.76) = [0.40], gam = 0.0195, Max-Change = 0.0019
Stage 3 = 21, CDLL = -19455.2, AR(0.76) = [0.40], gam = 0.0188, Max-Change = 0.0018
Stage 3 = 22, CDLL = -19447.6, AR(0.76) = [0.38], gam = 0.0181, Max-Change = 0.0016
Stage 3 = 23, CDLL = -19501.7, AR(0.76) = [0.40], gam = 0.0175, Max-Change = 0.0012
Stage 3 = 24, CDLL = -19573.5, AR(0.76) = [0.40], gam = 0.0169, Max-Change = 0.0022
Stage 3 = 25, CDLL = -19501.5, AR(0.76) = [0.39], gam = 0.0164, Max-Change = 0.0012
Stage 3 = 26, CDLL = -19462.3, AR(0.76) = [0.40], gam = 0.0159, Max-Change = 0.0022
Stage 3 = 27, CDLL = -19462.3, AR(0.76) = [0.39], gam = 0.0154, Max-Change = 0.0018
Stage 3 = 28, CDLL = -19462.4, AR(0.76) = [0.39], gam = 0.0150, Max-Change = 0.0016
Stage 3 = 29, CDLL = -19494.5, AR(0.76) = [0.40], gam = 0.0146, Max-Change = 0.0015
Stage 3 = 30, CDLL = -19507.6, AR(0.76) = [0.40], gam = 0.0142, Max-Change = 0.0008
Stage 3 = 31, CDLL = -19460.6, AR(0.76) = [0.40], gam = 0.0139, Max-Change = 0.0016
Stage 3 = 32, CDLL = -19501.1, AR(0.76) = [0.39], gam = 0.0135, Max-Change = 0.0012
Stage 3 = 33, CDLL = -19493.3, AR(0.76) = [0.39], gam = 0.0132, Max-Change = 0.0021
Stage 3 = 34, CDLL = -19416.1, AR(0.76) = [0.41], gam = 0.0129, Max-Change = 0.0013
Stage 3 = 35, CDLL = -19424.7, AR(0.76) = [0.38], gam = 0.0126, Max-Change = 0.0011
Stage 3 = 36, CDLL = -19424.3, AR(0.76) = [0.39], gam = 0.0124, Max-Change = 0.0014
Stage 3 = 37, CDLL = -19512.4, AR(0.76) = [0.41], gam = 0.0121, Max-Change = 0.0014
Stage 3 = 38, CDLL = -19520.4, AR(0.76) = [0.41], gam = 0.0119, Max-Change = 0.0016
Stage 3 = 39, CDLL = -19540.2, AR(0.76) = [0.40], gam = 0.0116, Max-Change = 0.0013
Stage 3 = 40, CDLL = -19510.7, AR(0.76) = [0.40], gam = 0.0114, Max-Change = 0.0026
Stage 3 = 41, CDLL = -19446.6, AR(0.76) = [0.41], gam = 0.0112, Max-Change = 0.0011
Stage 3 = 42, CDLL = -19483.8, AR(0.76) = [0.39], gam = 0.0110, Max-Change = 0.0009
Stage 3 = 43, CDLL = -19443.7, AR(0.76) = [0.38], gam = 0.0108, Max-Change = 0.0008
Stage 3 = 44, CDLL = -19552.5, AR(0.76) = [0.40], gam = 0.0106, Max-Change = 0.0013
Stage 3 = 45, CDLL = -19553.2, AR(0.76) = [0.40], gam = 0.0104, Max-Change = 0.0011
Stage 3 = 46, CDLL = -19498.5, AR(0.76) = [0.39], gam = 0.0102, Max-Change = 0.0009
Stage 3 = 47, CDLL = -19498.0, AR(0.76) = [0.38], gam = 0.0101, Max-Change = 0.0010
Stage 3 = 48, CDLL = -19534.3, AR(0.76) = [0.39], gam = 0.0099, Max-Change = 0.0009

#> 
#> Calculating log-likelihood...
coef(mod3)
#> $Item_1
#>        a1    a2 a3      d g u
#> par 0.881 1.526  0 -0.976 0 1
#> 
#> $Item_2
#>        a1    a2 a3     d g u
#> par 0.309 0.356  0 -1.52 0 1
#> 
#> $Item_3
#>        a1    a2 a3     d g u
#> par 0.507 0.739  0 1.522 0 1
#> 
#> $Item_4
#>        a1    a2 a3     d g u
#> par 1.374 0.753  0 0.101 0 1
#> 
#> $Item_5
#>        a1 a2    a3    d1    d2    d3
#> par 0.958  0 1.054 2.832 1.897 -0.51
#> 
#> $Item_6
#>        a1 a2    a3    d1    d2     d3
#> par 0.563  0 0.282 2.494 1.008 -1.043
#> 
#> $Item_7
#>        a1 a2    a3    d1     d2
#> par 0.817  0 0.604 2.002 -0.021
#> 
#> $Item_8
#>        a1 a2    a3     d g u
#> par 0.488  0 1.106 1.036 0 1
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 MEAN_3 COV_11 COV_21 COV_31 COV_22 COV_32 COV_33
#> par      0      0      0      1      0      0      1      0      1
#> 
summary(mod3)
#>            G    F1    F2     h2
#> Item_1 0.359 0.623       0.5173
#> Item_2 0.175 0.202       0.0714
#> Item_3 0.263 0.384       0.2169
#> Item_4 0.594 0.326       0.4587
#> Item_5 0.432       0.475 0.4120
#> Item_6 0.310       0.155 0.1204
#> Item_7 0.412       0.305 0.2628
#> Item_8 0.234       0.530 0.3352
#> 
#> SS loadings:  1.089 0.682 0.623 
#> Proportion Var:  0.136 0.085 0.078 
#> 
#> Factor correlations: 
#> 
#>    G F1 F2
#> G  1      
#> F1 0  1   
#> F2 0  0  1
residuals(mod3)
#> LD matrix (lower triangle) and standardized residual correlations (upper triangle)
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.046  -0.030  -0.007  -0.006   0.011   0.049 
#> 
#>        Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8
#> Item_1         0.007 -0.007  0.004 -0.007  0.049 -0.023 -0.010
#> Item_2  0.090         0.017 -0.025 -0.046  0.036  0.006  0.012
#> Item_3  0.088  0.563        -0.005 -0.034  0.014 -0.033 -0.012
#> Item_4  0.025  1.224  0.045         0.035 -0.029 -0.042  0.004
#> Item_5  0.112  4.234  2.273  2.392         0.031 -0.039 -0.036
#> Item_6  4.748  2.601  0.417  1.653  5.653        -0.034  0.011
#> Item_7  1.037  0.064  2.131  3.498  6.233  4.532        -0.015
#> Item_8  0.211  0.274  0.306  0.026  2.560  0.231  0.426       
anova(mod1,mod3)
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> mod1 24903.92 24959.67 24951.22 25032.74 -12428.96                
#> mod3 24904.18 24974.47 24963.82 25066.61 -12423.09 11.736  6 0.068

#####
# polynomial/combinations
data(SAT12)
data <- key2binary(SAT12,
                  key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))

model.quad <- '
       F1 = 1-32
  (F1*F1) = 1-32'


model.combo <- '
       F1 = 1-16
       F2 = 17-32
  (F1*F2) = 1-8'

(mod.quad <- mirt(data, model.quad))
#> 
Iteration: 1, Log-Lik: -10104.844, Max-Change: 2.31942
Iteration: 2, Log-Lik: -9572.707, Max-Change: 0.95981
Iteration: 3, Log-Lik: -9512.399, Max-Change: 0.74584
Iteration: 4, Log-Lik: -9495.470, Max-Change: 0.57352
Iteration: 5, Log-Lik: -9487.734, Max-Change: 0.38068
Iteration: 6, Log-Lik: -9482.541, Max-Change: 0.25814
Iteration: 7, Log-Lik: -9477.848, Max-Change: 0.19174
Iteration: 8, Log-Lik: -9473.207, Max-Change: 0.16400
Iteration: 9, Log-Lik: -9468.837, Max-Change: 0.14249
Iteration: 10, Log-Lik: -9458.312, Max-Change: 0.17409
Iteration: 11, Log-Lik: -9454.940, Max-Change: 0.11553
Iteration: 12, Log-Lik: -9452.934, Max-Change: 0.11647
Iteration: 13, Log-Lik: -9447.607, Max-Change: 0.14066
Iteration: 14, Log-Lik: -9446.061, Max-Change: 0.14140
Iteration: 15, Log-Lik: -9444.751, Max-Change: 0.14105
Iteration: 16, Log-Lik: -9438.230, Max-Change: 0.18679
Iteration: 17, Log-Lik: -9436.418, Max-Change: 0.17037
Iteration: 18, Log-Lik: -9435.137, Max-Change: 0.17097
Iteration: 19, Log-Lik: -9430.716, Max-Change: 0.11338
Iteration: 20, Log-Lik: -9430.014, Max-Change: 0.14199
Iteration: 21, Log-Lik: -9429.752, Max-Change: 0.17869
Iteration: 22, Log-Lik: -9429.315, Max-Change: 0.07197
Iteration: 23, Log-Lik: -9429.206, Max-Change: 0.08170
Iteration: 24, Log-Lik: -9429.111, Max-Change: 0.08251
Iteration: 25, Log-Lik: -9428.667, Max-Change: 0.08779
Iteration: 26, Log-Lik: -9428.607, Max-Change: 0.04593
Iteration: 27, Log-Lik: -9428.552, Max-Change: 0.04486
Iteration: 28, Log-Lik: -9428.249, Max-Change: 0.05193
Iteration: 29, Log-Lik: -9428.196, Max-Change: 0.04799
Iteration: 30, Log-Lik: -9428.146, Max-Change: 0.04650
Iteration: 31, Log-Lik: -9427.856, Max-Change: 0.04026
Iteration: 32, Log-Lik: -9427.809, Max-Change: 0.04062
Iteration: 33, Log-Lik: -9427.763, Max-Change: 0.04160
Iteration: 34, Log-Lik: -9427.494, Max-Change: 0.04167
Iteration: 35, Log-Lik: -9427.450, Max-Change: 0.04122
Iteration: 36, Log-Lik: -9427.406, Max-Change: 0.04257
Iteration: 37, Log-Lik: -9427.152, Max-Change: 0.04010
Iteration: 38, Log-Lik: -9427.110, Max-Change: 0.03609
Iteration: 39, Log-Lik: -9427.068, Max-Change: 0.03550
Iteration: 40, Log-Lik: -9426.819, Max-Change: 0.03305
Iteration: 41, Log-Lik: -9426.778, Max-Change: 0.03169
Iteration: 42, Log-Lik: -9426.737, Max-Change: 0.03190
Iteration: 43, Log-Lik: -9426.497, Max-Change: 0.03230
Iteration: 44, Log-Lik: -9426.456, Max-Change: 0.02954
Iteration: 45, Log-Lik: -9426.417, Max-Change: 0.02927
Iteration: 46, Log-Lik: -9426.186, Max-Change: 0.02865
Iteration: 47, Log-Lik: -9426.147, Max-Change: 0.02639
Iteration: 48, Log-Lik: -9426.109, Max-Change: 0.02626
Iteration: 49, Log-Lik: -9425.889, Max-Change: 0.02726
Iteration: 50, Log-Lik: -9425.853, Max-Change: 0.01939
Iteration: 51, Log-Lik: -9425.820, Max-Change: 0.02660
Iteration: 52, Log-Lik: -9425.753, Max-Change: 0.02701
Iteration: 53, Log-Lik: -9425.719, Max-Change: 0.02703
Iteration: 54, Log-Lik: -9425.686, Max-Change: 0.02701
Iteration: 55, Log-Lik: -9425.497, Max-Change: 0.02757
Iteration: 56, Log-Lik: -9425.466, Max-Change: 0.01835
Iteration: 57, Log-Lik: -9425.441, Max-Change: 0.02493
Iteration: 58, Log-Lik: -9425.389, Max-Change: 0.01742
Iteration: 59, Log-Lik: -9425.365, Max-Change: 0.02595
Iteration: 60, Log-Lik: -9425.339, Max-Change: 0.01680
Iteration: 61, Log-Lik: -9425.310, Max-Change: 0.02693
Iteration: 62, Log-Lik: -9425.285, Max-Change: 0.02657
Iteration: 63, Log-Lik: -9425.260, Max-Change: 0.01701
Iteration: 64, Log-Lik: -9425.233, Max-Change: 0.02793
Iteration: 65, Log-Lik: -9425.210, Max-Change: 0.02895
Iteration: 66, Log-Lik: -9425.188, Max-Change: 0.01635
Iteration: 67, Log-Lik: -9425.167, Max-Change: 0.02819
Iteration: 68, Log-Lik: -9425.145, Max-Change: 0.01593
Iteration: 69, Log-Lik: -9425.125, Max-Change: 0.02892
Iteration: 70, Log-Lik: -9425.089, Max-Change: 0.01502
Iteration: 71, Log-Lik: -9425.069, Max-Change: 0.01603
Iteration: 72, Log-Lik: -9425.049, Max-Change: 0.02470
Iteration: 73, Log-Lik: -9425.010, Max-Change: 0.01472
Iteration: 74, Log-Lik: -9424.993, Max-Change: 0.03242
Iteration: 75, Log-Lik: -9424.975, Max-Change: 0.01408
Iteration: 76, Log-Lik: -9424.962, Max-Change: 0.03166
Iteration: 77, Log-Lik: -9424.944, Max-Change: 0.01423
Iteration: 78, Log-Lik: -9424.927, Max-Change: 0.02611
Iteration: 79, Log-Lik: -9424.899, Max-Change: 0.01301
Iteration: 80, Log-Lik: -9424.884, Max-Change: 0.01370
Iteration: 81, Log-Lik: -9424.869, Max-Change: 0.03259
Iteration: 82, Log-Lik: -9424.842, Max-Change: 0.01352
Iteration: 83, Log-Lik: -9424.828, Max-Change: 0.01402
Iteration: 84, Log-Lik: -9424.814, Max-Change: 0.02744
Iteration: 85, Log-Lik: -9424.787, Max-Change: 0.01019
Iteration: 86, Log-Lik: -9424.775, Max-Change: 0.03497
Iteration: 87, Log-Lik: -9424.761, Max-Change: 0.01274
Iteration: 88, Log-Lik: -9424.752, Max-Change: 0.01236
Iteration: 89, Log-Lik: -9424.740, Max-Change: 0.02877
Iteration: 90, Log-Lik: -9424.727, Max-Change: 0.01094
Iteration: 91, Log-Lik: -9424.718, Max-Change: 0.01207
Iteration: 92, Log-Lik: -9424.707, Max-Change: 0.02948
Iteration: 93, Log-Lik: -9424.695, Max-Change: 0.00958
Iteration: 94, Log-Lik: -9424.688, Max-Change: 0.03017
Iteration: 95, Log-Lik: -9424.676, Max-Change: 0.00950
Iteration: 96, Log-Lik: -9424.667, Max-Change: 0.03043
Iteration: 97, Log-Lik: -9424.651, Max-Change: 0.00936
Iteration: 98, Log-Lik: -9424.642, Max-Change: 0.01048
Iteration: 99, Log-Lik: -9424.632, Max-Change: 0.01144
Iteration: 100, Log-Lik: -9424.582, Max-Change: 0.03020
Iteration: 101, Log-Lik: -9424.571, Max-Change: 0.03095
Iteration: 102, Log-Lik: -9424.561, Max-Change: 0.00739
Iteration: 103, Log-Lik: -9424.557, Max-Change: 0.04433
Iteration: 104, Log-Lik: -9424.547, Max-Change: 0.00803
Iteration: 105, Log-Lik: -9424.540, Max-Change: 0.00855
Iteration: 106, Log-Lik: -9424.502, Max-Change: 0.02970
Iteration: 107, Log-Lik: -9424.492, Max-Change: 0.03563
Iteration: 108, Log-Lik: -9424.484, Max-Change: 0.00841
Iteration: 109, Log-Lik: -9424.481, Max-Change: 0.00814
Iteration: 110, Log-Lik: -9424.475, Max-Change: 0.01042
Iteration: 111, Log-Lik: -9424.469, Max-Change: 0.03429
Iteration: 112, Log-Lik: -9424.458, Max-Change: 0.00609
Iteration: 113, Log-Lik: -9424.453, Max-Change: 0.00816
Iteration: 114, Log-Lik: -9424.448, Max-Change: 0.00888
Iteration: 115, Log-Lik: -9424.419, Max-Change: 0.00246
Iteration: 116, Log-Lik: -9424.415, Max-Change: 0.03113
Iteration: 117, Log-Lik: -9424.408, Max-Change: 0.00837
Iteration: 118, Log-Lik: -9424.406, Max-Change: 0.00928
Iteration: 119, Log-Lik: -9424.402, Max-Change: 0.00263
Iteration: 120, Log-Lik: -9424.399, Max-Change: 0.03641
Iteration: 121, Log-Lik: -9424.393, Max-Change: 0.00706
Iteration: 122, Log-Lik: -9424.389, Max-Change: 0.00935
Iteration: 123, Log-Lik: -9424.385, Max-Change: 0.00238
Iteration: 124, Log-Lik: -9424.383, Max-Change: 0.01117
Iteration: 125, Log-Lik: -9424.379, Max-Change: 0.00209
Iteration: 126, Log-Lik: -9424.376, Max-Change: 0.00902
Iteration: 127, Log-Lik: -9424.372, Max-Change: 0.01228
Iteration: 128, Log-Lik: -9424.368, Max-Change: 0.03758
Iteration: 129, Log-Lik: -9424.363, Max-Change: 0.00184
Iteration: 130, Log-Lik: -9424.363, Max-Change: 0.00698
Iteration: 131, Log-Lik: -9424.360, Max-Change: 0.00183
Iteration: 132, Log-Lik: -9424.358, Max-Change: 0.00923
Iteration: 133, Log-Lik: -9424.353, Max-Change: 0.00179
Iteration: 134, Log-Lik: -9424.352, Max-Change: 0.00987
Iteration: 135, Log-Lik: -9424.348, Max-Change: 0.00172
Iteration: 136, Log-Lik: -9424.348, Max-Change: 0.01123
Iteration: 137, Log-Lik: -9424.344, Max-Change: 0.00189
Iteration: 138, Log-Lik: -9424.343, Max-Change: 0.03765
Iteration: 139, Log-Lik: -9424.337, Max-Change: 0.00162
Iteration: 140, Log-Lik: -9424.336, Max-Change: 0.00582
Iteration: 141, Log-Lik: -9424.333, Max-Change: 0.00180
Iteration: 142, Log-Lik: -9424.332, Max-Change: 0.00186
Iteration: 143, Log-Lik: -9424.331, Max-Change: 0.00939
Iteration: 144, Log-Lik: -9424.328, Max-Change: 0.00177
Iteration: 145, Log-Lik: -9424.327, Max-Change: 0.01032
Iteration: 146, Log-Lik: -9424.324, Max-Change: 0.00170
Iteration: 147, Log-Lik: -9424.323, Max-Change: 0.01276
Iteration: 148, Log-Lik: -9424.319, Max-Change: 0.00158
Iteration: 149, Log-Lik: -9424.318, Max-Change: 0.03912
Iteration: 150, Log-Lik: -9424.314, Max-Change: 0.00163
Iteration: 151, Log-Lik: -9424.313, Max-Change: 0.00162
Iteration: 152, Log-Lik: -9424.312, Max-Change: 0.00169
Iteration: 153, Log-Lik: -9424.311, Max-Change: 0.00626
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Iteration: 469, Log-Lik: -9424.252, Max-Change: 0.00032
Iteration: 470, Log-Lik: -9424.252, Max-Change: 0.00058
Iteration: 471, Log-Lik: -9424.252, Max-Change: 0.00025
Iteration: 472, Log-Lik: -9424.252, Max-Change: 0.00020
Iteration: 473, Log-Lik: -9424.252, Max-Change: 0.00041
Iteration: 474, Log-Lik: -9424.251, Max-Change: 0.00082
Iteration: 475, Log-Lik: -9424.251, Max-Change: 0.00031
Iteration: 476, Log-Lik: -9424.251, Max-Change: 0.00057
Iteration: 477, Log-Lik: -9424.251, Max-Change: 0.00025
Iteration: 478, Log-Lik: -9424.251, Max-Change: 0.00020
Iteration: 479, Log-Lik: -9424.251, Max-Change: 0.00041
Iteration: 480, Log-Lik: -9424.251, Max-Change: 0.00080
Iteration: 481, Log-Lik: -9424.251, Max-Change: 0.00031
Iteration: 482, Log-Lik: -9424.251, Max-Change: 0.00057
Iteration: 483, Log-Lik: -9424.251, Max-Change: 0.00025
Iteration: 484, Log-Lik: -9424.251, Max-Change: 0.00020
Iteration: 485, Log-Lik: -9424.251, Max-Change: 0.00041
Iteration: 486, Log-Lik: -9424.251, Max-Change: 0.00081
Iteration: 487, Log-Lik: -9424.251, Max-Change: 0.00031
Iteration: 488, Log-Lik: -9424.251, Max-Change: 0.00056
Iteration: 489, Log-Lik: -9424.251, Max-Change: 0.00025
Iteration: 490, Log-Lik: -9424.251, Max-Change: 0.00020
Iteration: 491, Log-Lik: -9424.251, Max-Change: 0.00042
Iteration: 492, Log-Lik: -9424.251, Max-Change: 0.00080
Iteration: 493, Log-Lik: -9424.251, Max-Change: 0.00031
Iteration: 494, Log-Lik: -9424.250, Max-Change: 0.00057
Iteration: 495, Log-Lik: -9424.250, Max-Change: 0.00025
Iteration: 496, Log-Lik: -9424.250, Max-Change: 0.00020
Iteration: 497, Log-Lik: -9424.250, Max-Change: 0.00042
Iteration: 498, Log-Lik: -9424.250, Max-Change: 0.00080
Iteration: 499, Log-Lik: -9424.250, Max-Change: 0.00031
Iteration: 500, Log-Lik: -9424.250, Max-Change: 0.00056
#> Warning: EM cycles terminated after 500 iterations.
#> 
#> Call:
#> mirt(data = data, model = model.quad)
#> 
#> Full-information item factor analysis with 1 factor(s).
#> FAILED TO CONVERGE within 1e-04 tolerance after 500 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -9424.25
#> Estimated parameters: 96 
#> AIC = 19040.5
#> BIC = 19462.61; SABIC = 19157.83
#> 
summary(mod.quad)
#>               F1 (F1*F1)     h2
#> Item.1   0.24781  0.3206 0.1642
#> Item.2   0.31819  0.6617 0.5390
#> Item.3   0.18997  0.4629 0.2504
#> Item.4   0.22615  0.2802 0.1297
#> Item.5   0.27007  0.4764 0.2999
#> Item.6   0.23129  0.4340 0.2418
#> Item.7  -0.23295  0.6835 0.5215
#> Item.8   0.07142  0.3230 0.1094
#> Item.9   0.07120  0.2448 0.0650
#> Item.10  0.12985  0.4483 0.2178
#> Item.11  0.00489  0.9832 0.9667
#> Item.12  0.13130  0.0666 0.0217
#> Item.13 -0.12244  0.6412 0.4261
#> Item.14  0.42256  0.5436 0.4741
#> Item.15 -0.25904  0.8074 0.7190
#> Item.16  0.15771  0.3584 0.1533
#> Item.17 -0.30637  0.8837 0.8747
#> Item.18  0.22597  0.6542 0.4790
#> Item.19  0.17275  0.4037 0.1928
#> Item.20  0.37087  0.7925 0.7656
#> Item.21 -0.36576  0.5739 0.4631
#> Item.22 -0.27360  0.9330 0.9454
#> Item.23  0.41421  0.2192 0.2196
#> Item.24 -0.13434  0.7637 0.6012
#> Item.25  0.60732  0.2554 0.4341
#> Item.26  0.35413  0.6293 0.5214
#> Item.27 -0.05234  0.9284 0.8647
#> Item.28  0.09058  0.5130 0.2714
#> Item.29  0.26616  0.3615 0.2016
#> Item.30  0.05551  0.1696 0.0319
#> Item.31  0.25711  0.9259 0.9234
#> Item.32  0.01322  0.1086 0.0120
#> 
#> SS loadings:  2.116 10.985 
#> Proportion Var:  0.066 0.343 
#> 
#> Factor correlations: 
#> 
#>    F1
#> F1  1
(mod.combo <- mirt(data, model.combo))
#> 
Iteration: 1, Log-Lik: -9815.422, Max-Change: 0.55645
Iteration: 2, Log-Lik: -9678.421, Max-Change: 0.55710
Iteration: 3, Log-Lik: -9643.201, Max-Change: 0.22294
Iteration: 4, Log-Lik: -9629.356, Max-Change: 0.13823
Iteration: 5, Log-Lik: -9624.018, Max-Change: 0.09346
Iteration: 6, Log-Lik: -9621.772, Max-Change: 0.05784
Iteration: 7, Log-Lik: -9620.830, Max-Change: 0.04210
Iteration: 8, Log-Lik: -9620.410, Max-Change: 0.03312
Iteration: 9, Log-Lik: -9620.193, Max-Change: 0.02684
Iteration: 10, Log-Lik: -9619.955, Max-Change: 0.01173
Iteration: 11, Log-Lik: -9619.926, Max-Change: 0.00961
Iteration: 12, Log-Lik: -9619.907, Max-Change: 0.00749
Iteration: 13, Log-Lik: -9619.875, Max-Change: 0.00255
Iteration: 14, Log-Lik: -9619.874, Max-Change: 0.00194
Iteration: 15, Log-Lik: -9619.873, Max-Change: 0.00165
Iteration: 16, Log-Lik: -9619.871, Max-Change: 0.00084
Iteration: 17, Log-Lik: -9619.871, Max-Change: 0.00022
Iteration: 18, Log-Lik: -9619.871, Max-Change: 0.00011
Iteration: 19, Log-Lik: -9619.871, Max-Change: 0.00046
Iteration: 20, Log-Lik: -9619.871, Max-Change: 0.00039
Iteration: 21, Log-Lik: -9619.871, Max-Change: 0.00014
Iteration: 22, Log-Lik: -9619.871, Max-Change: 0.00010
#> 
#> Call:
#> mirt(data = data, model = model.combo)
#> 
#> Full-information item factor analysis with 2 factor(s).
#> Converged within 1e-04 tolerance after 22 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 31
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -9619.871
#> Estimated parameters: 72 
#> AIC = 19383.74
#> BIC = 19700.32; SABIC = 19471.74
#> 
anova(mod.combo, mod.quad)
#>                AIC    SABIC       HQ      BIC    logLik      X2 df p
#> mod.combo 19383.74 19471.74 19506.98 19700.32 -9619.871             
#> mod.quad  19040.50 19157.83 19204.82 19462.60 -9424.250 391.241 24 0

# non-linear item and test plots
plot(mod.quad)

plot(mod.combo, type = 'SE')

itemplot(mod.quad, 1, type = 'score')

itemplot(mod.combo, 2, type = 'score')

itemplot(mod.combo, 2, type = 'infocontour')


## empirical histogram examples (normal, skew and bimodality)
# make some data
set.seed(1234)
a <- matrix(rlnorm(50, .2, .2))
d <- matrix(rnorm(50))
ThetaNormal <- matrix(rnorm(2000))
ThetaBimodal <- scale(matrix(c(rnorm(1000, -2), rnorm(1000,2)))) #bimodal
ThetaSkew <- scale(matrix(rchisq(2000, 3))) #positive skew
datNormal <- simdata(a, d, 2000, itemtype = '2PL', Theta=ThetaNormal)
datBimodal <- simdata(a, d, 2000, itemtype = '2PL', Theta=ThetaBimodal)
datSkew <- simdata(a, d, 2000, itemtype = '2PL', Theta=ThetaSkew)

normal <- mirt(datNormal, 1, dentype = "empiricalhist")
#> 
Iteration: 1, Log-Lik: -54606.961, Max-Change: 0.52637
Iteration: 2, Log-Lik: -54260.672, Max-Change: 0.07687
Iteration: 3, Log-Lik: -54251.710, Max-Change: 0.02534
Iteration: 4, Log-Lik: -54249.817, Max-Change: 0.00602
Iteration: 5, Log-Lik: -54249.069, Max-Change: 0.00358
Iteration: 6, Log-Lik: -54248.637, Max-Change: 0.00298
Iteration: 7, Log-Lik: -54248.348, Max-Change: 0.00222
Iteration: 8, Log-Lik: -54248.113, Max-Change: 0.00045
Iteration: 9, Log-Lik: -54247.934, Max-Change: 0.00033
Iteration: 10, Log-Lik: -54247.783, Max-Change: 0.00018
Iteration: 11, Log-Lik: -54247.651, Max-Change: 0.00025
Iteration: 12, Log-Lik: -54247.534, Max-Change: 0.00028
Iteration: 13, Log-Lik: -54247.428, Max-Change: 0.00038
Iteration: 14, Log-Lik: -54247.329, Max-Change: 0.00015
Iteration: 15, Log-Lik: -54247.239, Max-Change: 0.00010
Iteration: 16, Log-Lik: -54247.154, Max-Change: 0.00013
Iteration: 17, Log-Lik: -54247.075, Max-Change: 0.00015
Iteration: 18, Log-Lik: -54247.000, Max-Change: 0.00015
Iteration: 19, Log-Lik: -54246.929, Max-Change: 0.00014
Iteration: 20, Log-Lik: -54246.861, Max-Change: 0.00007
plot(normal, type = 'empiricalhist')

histogram(ThetaNormal, breaks=30)


bimodal <- mirt(datBimodal, 1, dentype = "empiricalhist")
#> 
Iteration: 1, Log-Lik: -54051.451, Max-Change: 0.73865
Iteration: 2, Log-Lik: -53444.408, Max-Change: 0.07607
Iteration: 3, Log-Lik: -53418.953, Max-Change: 0.01449
Iteration: 4, Log-Lik: -53412.710, Max-Change: 0.00800
Iteration: 5, Log-Lik: -53410.613, Max-Change: 0.00559
Iteration: 6, Log-Lik: -53409.823, Max-Change: 0.00286
Iteration: 7, Log-Lik: -53409.485, Max-Change: 0.00174
Iteration: 8, Log-Lik: -53409.317, Max-Change: 0.00111
Iteration: 9, Log-Lik: -53409.216, Max-Change: 0.00036
Iteration: 10, Log-Lik: -53409.147, Max-Change: 0.00039
Iteration: 11, Log-Lik: -53409.095, Max-Change: 0.00034
Iteration: 12, Log-Lik: -53409.052, Max-Change: 0.00029
Iteration: 13, Log-Lik: -53409.016, Max-Change: 0.00021
Iteration: 14, Log-Lik: -53408.985, Max-Change: 0.00011
Iteration: 15, Log-Lik: -53408.958, Max-Change: 0.00010
plot(bimodal, type = 'empiricalhist')

histogram(ThetaBimodal, breaks=30)


skew <- mirt(datSkew, 1, dentype = "empiricalhist")
#> 
Iteration: 1, Log-Lik: -55526.768, Max-Change: 0.46059
Iteration: 2, Log-Lik: -55135.282, Max-Change: 0.12740
Iteration: 3, Log-Lik: -55098.715, Max-Change: 0.03479
Iteration: 4, Log-Lik: -55087.259, Max-Change: 0.01382
Iteration: 5, Log-Lik: -55082.153, Max-Change: 0.00984
Iteration: 6, Log-Lik: -55079.400, Max-Change: 0.00644
Iteration: 7, Log-Lik: -55077.672, Max-Change: 0.00111
Iteration: 8, Log-Lik: -55076.535, Max-Change: 0.00184
Iteration: 9, Log-Lik: -55075.706, Max-Change: 0.00126
Iteration: 10, Log-Lik: -55075.062, Max-Change: 0.00190
Iteration: 11, Log-Lik: -55074.554, Max-Change: 0.00104
Iteration: 12, Log-Lik: -55074.138, Max-Change: 0.00074
Iteration: 13, Log-Lik: -55073.786, Max-Change: 0.00026
Iteration: 14, Log-Lik: -55073.490, Max-Change: 0.00057
Iteration: 15, Log-Lik: -55073.235, Max-Change: 0.00054
Iteration: 16, Log-Lik: -55073.009, Max-Change: 0.00119
Iteration: 17, Log-Lik: -55072.809, Max-Change: 0.00028
Iteration: 18, Log-Lik: -55072.635, Max-Change: 0.00029
Iteration: 19, Log-Lik: -55072.479, Max-Change: 0.00019
Iteration: 20, Log-Lik: -55072.338, Max-Change: 0.00025
Iteration: 21, Log-Lik: -55072.211, Max-Change: 0.00025
Iteration: 22, Log-Lik: -55072.091, Max-Change: 0.00062
Iteration: 23, Log-Lik: -55071.982, Max-Change: 0.00020
Iteration: 24, Log-Lik: -55071.882, Max-Change: 0.00021
Iteration: 25, Log-Lik: -55071.787, Max-Change: 0.00028
Iteration: 26, Log-Lik: -55071.699, Max-Change: 0.00019
Iteration: 27, Log-Lik: -55071.617, Max-Change: 0.00019
Iteration: 28, Log-Lik: -55071.538, Max-Change: 0.00033
Iteration: 29, Log-Lik: -55071.464, Max-Change: 0.00016
Iteration: 30, Log-Lik: -55071.394, Max-Change: 0.00017
Iteration: 31, Log-Lik: -55071.327, Max-Change: 0.00023
Iteration: 32, Log-Lik: -55071.263, Max-Change: 0.00015
Iteration: 33, Log-Lik: -55071.203, Max-Change: 0.00015
Iteration: 34, Log-Lik: -55071.144, Max-Change: 0.00018
Iteration: 35, Log-Lik: -55071.088, Max-Change: 0.00013
Iteration: 36, Log-Lik: -55071.035, Max-Change: 0.00014
Iteration: 37, Log-Lik: -55070.983, Max-Change: 0.00014
Iteration: 38, Log-Lik: -55070.933, Max-Change: 0.00012
Iteration: 39, Log-Lik: -55070.885, Max-Change: 0.00012
Iteration: 40, Log-Lik: -55070.838, Max-Change: 0.00011
Iteration: 41, Log-Lik: -55070.793, Max-Change: 0.00010
Iteration: 42, Log-Lik: -55070.750, Max-Change: 0.00011
Iteration: 43, Log-Lik: -55070.708, Max-Change: 0.00009
plot(skew, type = 'empiricalhist')

histogram(ThetaSkew, breaks=30)


#####
# non-linear parameter constraints with Rsolnp package (nloptr supported as well):
# Find Rasch model subject to the constraint that the intercepts sum to 0

dat <- expand.table(LSAT6)
itemstats(dat)
#> $overall
#>     N mean_total.score sd_total.score ave.r sd.r alpha SEM.alpha
#>  1000            3.819          1.035 0.077 0.03 0.295     0.869
#> 
#> $itemstats
#>           N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item_1 1000 0.924 0.265   0.362         0.113       0.275
#> Item_2 1000 0.709 0.454   0.567         0.153       0.238
#> Item_3 1000 0.553 0.497   0.618         0.173       0.217
#> Item_4 1000 0.763 0.425   0.534         0.144       0.246
#> Item_5 1000 0.870 0.336   0.435         0.122       0.266
#> 
#> $proportions
#>            0     1
#> Item_1 0.076 0.924
#> Item_2 0.291 0.709
#> Item_3 0.447 0.553
#> Item_4 0.237 0.763
#> Item_5 0.130 0.870
#> 

# free latent mean and variance terms
model <- 'Theta = 1-5
          MEAN = Theta
          COV = Theta*Theta'

# view how vector of parameters is organized internally
sv <- mirt(dat, model, itemtype = 'Rasch', pars = 'values')
sv[sv$est, ]
#>    group   item     class   name parnum value lbound ubound  est const nconst
#> 2    all Item_1      dich      d      2 2.815   -Inf    Inf TRUE  none   none
#> 6    all Item_2      dich      d      6 1.082   -Inf    Inf TRUE  none   none
#> 10   all Item_3      dich      d     10 0.262   -Inf    Inf TRUE  none   none
#> 14   all Item_4      dich      d     14 1.407   -Inf    Inf TRUE  none   none
#> 18   all Item_5      dich      d     18 2.214   -Inf    Inf TRUE  none   none
#> 21   all  GROUP GroupPars MEAN_1     21 0.000   -Inf    Inf TRUE  none   none
#> 22   all  GROUP GroupPars COV_11     22 1.000      0    Inf TRUE  none   none
#>    prior.type prior_1 prior_2
#> 2        none     NaN     NaN
#> 6        none     NaN     NaN
#> 10       none     NaN     NaN
#> 14       none     NaN     NaN
#> 18       none     NaN     NaN
#> 21       none     NaN     NaN
#> 22       none     NaN     NaN

# constraint: create function for solnp to compute constraint, and declare value in eqB
eqfun <- function(p, optim_args) sum(p[1:5]) #could use browser() here, if it helps
LB <- c(rep(-15, 6), 1e-4) # more reasonable lower bound for variance term

mod <- mirt(dat, model, sv=sv, itemtype = 'Rasch', optimizer = 'solnp',
   solnp_args=list(eqfun=eqfun, eqB=0, LB=LB))
#> 
Iteration: 1, Log-Lik: -2473.219, Max-Change: 2.01548
Iteration: 2, Log-Lik: -2978.093, Max-Change: 0.64564
Iteration: 3, Log-Lik: -2637.888, Max-Change: 0.30377
Iteration: 4, Log-Lik: -2536.573, Max-Change: 0.16978
Iteration: 5, Log-Lik: -2498.679, Max-Change: 0.10519
Iteration: 6, Log-Lik: -2482.438, Max-Change: 0.06941
Iteration: 7, Log-Lik: -2474.859, Max-Change: 0.04771
Iteration: 8, Log-Lik: -2471.120, Max-Change: 0.03371
Iteration: 9, Log-Lik: -2469.203, Max-Change: 0.02429
Iteration: 10, Log-Lik: -2468.200, Max-Change: 0.01951
Iteration: 11, Log-Lik: -2467.607, Max-Change: 0.01270
Iteration: 12, Log-Lik: -2467.329, Max-Change: 0.00946
Iteration: 13, Log-Lik: -2467.179, Max-Change: 0.00864
Iteration: 14, Log-Lik: -2467.073, Max-Change: 0.00542
Iteration: 15, Log-Lik: -2467.027, Max-Change: 0.00401
Iteration: 16, Log-Lik: -2467.001, Max-Change: 0.00343
Iteration: 17, Log-Lik: -2466.983, Max-Change: 0.00217
Iteration: 18, Log-Lik: -2466.975, Max-Change: 0.00166
Iteration: 19, Log-Lik: -2466.969, Max-Change: 0.00191
Iteration: 20, Log-Lik: -2466.964, Max-Change: 0.00105
Iteration: 21, Log-Lik: -2466.961, Max-Change: 0.00106
Iteration: 22, Log-Lik: -2466.958, Max-Change: 0.00095
Iteration: 23, Log-Lik: -2466.956, Max-Change: 0.00078
Iteration: 24, Log-Lik: -2466.954, Max-Change: 0.00112
Iteration: 25, Log-Lik: -2466.952, Max-Change: 0.00105
Iteration: 26, Log-Lik: -2466.950, Max-Change: 0.00099
Iteration: 27, Log-Lik: -2466.949, Max-Change: 0.00093
Iteration: 28, Log-Lik: -2466.947, Max-Change: 0.00087
Iteration: 29, Log-Lik: -2466.946, Max-Change: 0.00085
Iteration: 30, Log-Lik: -2466.945, Max-Change: 0.00079
Iteration: 31, Log-Lik: -2466.944, Max-Change: 0.00075
Iteration: 32, Log-Lik: -2466.944, Max-Change: 0.00020
Iteration: 33, Log-Lik: -2466.943, Max-Change: 0.00014
Iteration: 34, Log-Lik: -2466.943, Max-Change: 0.00010
print(mod)
#> 
#> Call:
#> mirt(data = dat, model = model, itemtype = "Rasch", optimizer = "solnp", 
#>     solnp_args = list(eqfun = eqfun, eqB = 0, LB = LB), sv = sv)
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 34 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: solnp 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -2466.943
#> Estimated parameters: 7 
#> AIC = 4947.887
#> BIC = 4982.241; SABIC = 4960.009
#> G2 (25) = 21.81, p = 0.6467
#> RMSEA = 0, CFI = NaN, TLI = NaN
coef(mod)
#> $Item_1
#>     a1     d g u
#> par  1 1.253 0 1
#> 
#> $Item_2
#>     a1      d g u
#> par  1 -0.475 0 1
#> 
#> $Item_3
#>     a1      d g u
#> par  1 -1.233 0 1
#> 
#> $Item_4
#>     a1      d g u
#> par  1 -0.168 0 1
#> 
#> $Item_5
#>     a1     d g u
#> par  1 0.623 0 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par  1.472  0.559
#> 
(ds <- sapply(coef(mod)[1:5], function(x) x[,'d']))
#>     Item_1     Item_2     Item_3     Item_4     Item_5 
#>  1.2529432 -0.4754429 -1.2327196 -0.1681687  0.6233879 
sum(ds)
#> [1] 4.635181e-15

# same likelihood location as: mirt(dat, 1, itemtype = 'Rasch')


#######
# latent regression Rasch model

# simulate data
set.seed(1234)
N <- 1000

# covariates
X1 <- rnorm(N); X2 <- rnorm(N)
covdata <- data.frame(X1, X2, X3 = rnorm(N))
Theta <- matrix(0.5 * X1 + -1 * X2 + rnorm(N, sd = 0.5))

# items and response data
a <- matrix(1, 20); d <- matrix(rnorm(20))
dat <- simdata(a, d, 1000, itemtype = '2PL', Theta=Theta)

# unconditional Rasch model
mod0 <- mirt(dat, 1, 'Rasch', SE=TRUE)
#> 
Iteration: 1, Log-Lik: -10962.384, Max-Change: 0.17087
Iteration: 2, Log-Lik: -10951.163, Max-Change: 0.10230
Iteration: 3, Log-Lik: -10947.935, Max-Change: 0.05720
Iteration: 4, Log-Lik: -10947.034, Max-Change: 0.03160
Iteration: 5, Log-Lik: -10946.813, Max-Change: 0.01521
Iteration: 6, Log-Lik: -10946.755, Max-Change: 0.00777
Iteration: 7, Log-Lik: -10946.739, Max-Change: 0.00404
Iteration: 8, Log-Lik: -10946.735, Max-Change: 0.00199
Iteration: 9, Log-Lik: -10946.733, Max-Change: 0.00101
Iteration: 10, Log-Lik: -10946.732, Max-Change: 0.00058
Iteration: 11, Log-Lik: -10946.732, Max-Change: 0.00029
Iteration: 12, Log-Lik: -10946.732, Max-Change: 0.00021
Iteration: 13, Log-Lik: -10946.731, Max-Change: 0.00013
Iteration: 14, Log-Lik: -10946.731, Max-Change: 0.00006
#> 
#> Calculating information matrix...
coef(mod0, printSE=TRUE)
#> $Item_1
#>     a1      d logit(g) logit(u)
#> par  1 -0.998     -999      999
#> SE  NA  0.085       NA       NA
#> 
#> $Item_2
#>     a1      d logit(g) logit(u)
#> par  1 -0.917     -999      999
#> SE  NA  0.085       NA       NA
#> 
#> $Item_3
#>     a1      d logit(g) logit(u)
#> par  1 -0.099     -999      999
#> SE  NA  0.081       NA       NA
#> 
#> $Item_4
#>     a1     d logit(g) logit(u)
#> par  1 1.893     -999      999
#> SE  NA 0.099       NA       NA
#> 
#> $Item_5
#>     a1     d logit(g) logit(u)
#> par  1 0.610     -999      999
#> SE  NA 0.082       NA       NA
#> 
#> $Item_6
#>     a1     d logit(g) logit(u)
#> par  1 1.071     -999      999
#> SE  NA 0.086       NA       NA
#> 
#> $Item_7
#>     a1      d logit(g) logit(u)
#> par  1 -0.074     -999      999
#> SE  NA  0.081       NA       NA
#> 
#> $Item_8
#>     a1      d logit(g) logit(u)
#> par  1 -1.405     -999      999
#> SE  NA  0.090       NA       NA
#> 
#> $Item_9
#>     a1     d logit(g) logit(u)
#> par  1 0.707     -999      999
#> SE  NA 0.083       NA       NA
#> 
#> $Item_10
#>     a1      d logit(g) logit(u)
#> par  1 -0.258     -999      999
#> SE  NA  0.081       NA       NA
#> 
#> $Item_11
#>     a1     d logit(g) logit(u)
#> par  1 0.336     -999      999
#> SE  NA 0.081       NA       NA
#> 
#> $Item_12
#>     a1     d logit(g) logit(u)
#> par  1 0.891     -999      999
#> SE  NA 0.084       NA       NA
#> 
#> $Item_13
#>     a1     d logit(g) logit(u)
#> par  1 0.653     -999      999
#> SE  NA 0.083       NA       NA
#> 
#> $Item_14
#>     a1      d logit(g) logit(u)
#> par  1 -1.942     -999      999
#> SE  NA  0.099       NA       NA
#> 
#> $Item_15
#>     a1      d logit(g) logit(u)
#> par  1 -2.143     -999      999
#> SE  NA  0.104       NA       NA
#> 
#> $Item_16
#>     a1     d logit(g) logit(u)
#> par  1 1.759     -999      999
#> SE  NA 0.096       NA       NA
#> 
#> $Item_17
#>     a1      d logit(g) logit(u)
#> par  1 -1.015     -999      999
#> SE  NA  0.085       NA       NA
#> 
#> $Item_18
#>     a1      d logit(g) logit(u)
#> par  1 -1.009     -999      999
#> SE  NA  0.085       NA       NA
#> 
#> $Item_19
#>     a1      d logit(g) logit(u)
#> par  1 -1.251     -999      999
#> SE  NA  0.088       NA       NA
#> 
#> $Item_20
#>     a1      d logit(g) logit(u)
#> par  1 -0.619     -999      999
#> SE  NA  0.082       NA       NA
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0  1.393
#> SE      NA  0.085
#> 

# conditional model using X1, X2, and X3 (bad) as predictors of Theta
mod1 <- mirt(dat, 1, 'Rasch', covdata=covdata, formula = ~ X1 + X2 + X3, SE=TRUE)
#> 
Iteration: 1, Log-Lik: -10962.384, Max-Change: 0.75355
Iteration: 2, Log-Lik: -10579.333, Max-Change: 0.51116
Iteration: 3, Log-Lik: -10430.132, Max-Change: 0.19355
Iteration: 4, Log-Lik: -10388.086, Max-Change: 0.08722
Iteration: 5, Log-Lik: -10372.328, Max-Change: 0.04911
Iteration: 6, Log-Lik: -10364.865, Max-Change: 0.03133
Iteration: 7, Log-Lik: -10360.851, Max-Change: 0.02159
Iteration: 8, Log-Lik: -10358.517, Max-Change: 0.01565
Iteration: 9, Log-Lik: -10357.088, Max-Change: 0.01176
Iteration: 10, Log-Lik: -10356.147, Max-Change: 0.00929
Iteration: 11, Log-Lik: -10355.550, Max-Change: 0.00722
Iteration: 12, Log-Lik: -10355.163, Max-Change: 0.00572
Iteration: 13, Log-Lik: -10354.904, Max-Change: 0.00468
Iteration: 14, Log-Lik: -10354.718, Max-Change: 0.00377
Iteration: 15, Log-Lik: -10354.597, Max-Change: 0.00307
Iteration: 16, Log-Lik: -10354.513, Max-Change: 0.00257
Iteration: 17, Log-Lik: -10354.451, Max-Change: 0.00211
Iteration: 18, Log-Lik: -10354.409, Max-Change: 0.00175
Iteration: 19, Log-Lik: -10354.380, Max-Change: 0.00148
Iteration: 20, Log-Lik: -10354.358, Max-Change: 0.00123
Iteration: 21, Log-Lik: -10354.344, Max-Change: 0.00103
Iteration: 22, Log-Lik: -10354.333, Max-Change: 0.00088
Iteration: 23, Log-Lik: -10354.325, Max-Change: 0.00074
Iteration: 24, Log-Lik: -10354.320, Max-Change: 0.00062
Iteration: 25, Log-Lik: -10354.316, Max-Change: 0.00053
Iteration: 26, Log-Lik: -10354.313, Max-Change: 0.00045
Iteration: 27, Log-Lik: -10354.311, Max-Change: 0.00038
Iteration: 28, Log-Lik: -10354.309, Max-Change: 0.00032
Iteration: 29, Log-Lik: -10354.308, Max-Change: 0.00027
Iteration: 30, Log-Lik: -10354.307, Max-Change: 0.00023
Iteration: 31, Log-Lik: -10354.307, Max-Change: 0.00028
Iteration: 32, Log-Lik: -10354.306, Max-Change: 0.00017
Iteration: 33, Log-Lik: -10354.306, Max-Change: 0.00014
Iteration: 34, Log-Lik: -10354.306, Max-Change: 0.00012
Iteration: 35, Log-Lik: -10354.306, Max-Change: 0.00010
Iteration: 36, Log-Lik: -10354.306, Max-Change: 0.00009
#> 
#> Calculating information matrix...
coef(mod1, printSE=TRUE)
#> $Item_1
#>     a1      d logit(g) logit(u)
#> par  1 -0.967     -999      999
#> SE  NA  0.078       NA       NA
#> 
#> $Item_2
#>     a1      d logit(g) logit(u)
#> par  1 -0.887     -999      999
#> SE  NA  0.077       NA       NA
#> 
#> $Item_3
#>     a1      d logit(g) logit(u)
#> par  1 -0.068     -999      999
#> SE  NA  0.073       NA       NA
#> 
#> $Item_4
#>     a1     d logit(g) logit(u)
#> par  1 1.920     -999      999
#> SE  NA 0.092       NA       NA
#> 
#> $Item_5
#>     a1     d logit(g) logit(u)
#> par  1 0.640     -999      999
#> SE  NA 0.075       NA       NA
#> 
#> $Item_6
#>     a1     d logit(g) logit(u)
#> par  1 1.100     -999      999
#> SE  NA 0.079       NA       NA
#> 
#> $Item_7
#>     a1      d logit(g) logit(u)
#> par  1 -0.043     -999      999
#> SE  NA  0.073       NA       NA
#> 
#> $Item_8
#>     a1      d logit(g) logit(u)
#> par  1 -1.375     -999      999
#> SE  NA  0.083       NA       NA
#> 
#> $Item_9
#>     a1     d logit(g) logit(u)
#> par  1 0.737     -999      999
#> SE  NA 0.076       NA       NA
#> 
#> $Item_10
#>     a1      d logit(g) logit(u)
#> par  1 -0.227     -999      999
#> SE  NA  0.073       NA       NA
#> 
#> $Item_11
#>     a1     d logit(g) logit(u)
#> par  1 0.367     -999      999
#> SE  NA 0.074       NA       NA
#> 
#> $Item_12
#>     a1     d logit(g) logit(u)
#> par  1 0.921     -999      999
#> SE  NA 0.077       NA       NA
#> 
#> $Item_13
#>     a1     d logit(g) logit(u)
#> par  1 0.683     -999      999
#> SE  NA 0.075       NA       NA
#> 
#> $Item_14
#>     a1      d logit(g) logit(u)
#> par  1 -1.913     -999      999
#> SE  NA  0.093       NA       NA
#> 
#> $Item_15
#>     a1      d logit(g) logit(u)
#> par  1 -2.114     -999      999
#> SE  NA  0.098       NA       NA
#> 
#> $Item_16
#>     a1     d logit(g) logit(u)
#> par  1 1.786     -999      999
#> SE  NA 0.090       NA       NA
#> 
#> $Item_17
#>     a1      d logit(g) logit(u)
#> par  1 -0.985     -999      999
#> SE  NA  0.078       NA       NA
#> 
#> $Item_18
#>     a1      d logit(g) logit(u)
#> par  1 -0.979     -999      999
#> SE  NA  0.078       NA       NA
#> 
#> $Item_19
#>     a1      d logit(g) logit(u)
#> par  1 -1.221     -999      999
#> SE  NA  0.081       NA       NA
#> 
#> $Item_20
#>     a1      d logit(g) logit(u)
#> par  1 -0.589     -999      999
#> SE  NA  0.075       NA       NA
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0  0.210
#> SE      NA  0.011
#> 
#> $lr.betas
#> $lr.betas$betas
#>                 F1
#> (Intercept)  0.000
#> X1           0.513
#> X2          -1.003
#> X3          -0.003
#> 
#> $lr.betas$SE
#>                F1
#> (Intercept)    NA
#> X1          0.015
#> X2          0.015
#> X3          0.014
#> 
#> 
coef(mod1, simplify=TRUE)
#> $items
#>         a1      d g u
#> Item_1   1 -0.967 0 1
#> Item_2   1 -0.887 0 1
#> Item_3   1 -0.068 0 1
#> Item_4   1  1.920 0 1
#> Item_5   1  0.640 0 1
#> Item_6   1  1.100 0 1
#> Item_7   1 -0.043 0 1
#> Item_8   1 -1.375 0 1
#> Item_9   1  0.737 0 1
#> Item_10  1 -0.227 0 1
#> Item_11  1  0.367 0 1
#> Item_12  1  0.921 0 1
#> Item_13  1  0.683 0 1
#> Item_14  1 -1.913 0 1
#> Item_15  1 -2.114 0 1
#> Item_16  1  1.786 0 1
#> Item_17  1 -0.985 0 1
#> Item_18  1 -0.979 0 1
#> Item_19  1 -1.221 0 1
#> Item_20  1 -0.589 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>      F1
#> F1 0.21
#> 
#> $lr.betas
#> $lr.betas$betas
#>                 F1
#> (Intercept)  0.000
#> X1           0.513
#> X2          -1.003
#> X3          -0.003
#> 
#> $lr.betas$CI_2.5
#>                 F1
#> (Intercept)     NA
#> X1           0.485
#> X2          -1.032
#> X3          -0.031
#> 
#> $lr.betas$CI_97.5
#>                 F1
#> (Intercept)     NA
#> X1           0.542
#> X2          -0.974
#> X3           0.025
#> 
#> 
anova(mod0, mod1)  # jointly significant predictors of theta
#>           AIC    SABIC       HQ      BIC    logLik       X2 df p
#> mod0 21935.46 21971.83 21974.63 22038.53 -10946.73              
#> mod1 20756.61 20798.17 20801.38 20874.40 -10354.31 1184.851  3 0

# large sample z-ratios and p-values (if one cares)
cfs <- coef(mod1, printSE=TRUE)
(z <- cfs$lr.betas[[1]] / cfs$lr.betas[[2]])
#>                      F1
#> (Intercept)          NA
#> X1           35.2668946
#> X2          -67.5847949
#> X3           -0.2114561
round(pnorm(abs(z[,1]), lower.tail=FALSE)*2, 3)
#> (Intercept)          X1          X2          X3 
#>          NA       0.000       0.000       0.833 

# drop predictor for nested comparison
mod1b <- mirt(dat, 1, 'Rasch', covdata=covdata, formula = ~ X1 + X2)
#> 
Iteration: 1, Log-Lik: -10962.384, Max-Change: 0.75353
Iteration: 2, Log-Lik: -10579.333, Max-Change: 0.51116
Iteration: 3, Log-Lik: -10430.135, Max-Change: 0.19355
Iteration: 4, Log-Lik: -10388.092, Max-Change: 0.08722
Iteration: 5, Log-Lik: -10372.335, Max-Change: 0.04910
Iteration: 6, Log-Lik: -10364.873, Max-Change: 0.03133
Iteration: 7, Log-Lik: -10360.859, Max-Change: 0.02159
Iteration: 8, Log-Lik: -10358.525, Max-Change: 0.01565
Iteration: 9, Log-Lik: -10357.097, Max-Change: 0.01176
Iteration: 10, Log-Lik: -10356.155, Max-Change: 0.00929
Iteration: 11, Log-Lik: -10355.559, Max-Change: 0.00722
Iteration: 12, Log-Lik: -10355.172, Max-Change: 0.00572
Iteration: 13, Log-Lik: -10354.912, Max-Change: 0.00468
Iteration: 14, Log-Lik: -10354.727, Max-Change: 0.00377
Iteration: 15, Log-Lik: -10354.606, Max-Change: 0.00307
Iteration: 16, Log-Lik: -10354.522, Max-Change: 0.00257
Iteration: 17, Log-Lik: -10354.460, Max-Change: 0.00211
Iteration: 18, Log-Lik: -10354.418, Max-Change: 0.00175
Iteration: 19, Log-Lik: -10354.389, Max-Change: 0.00148
Iteration: 20, Log-Lik: -10354.368, Max-Change: 0.00123
Iteration: 21, Log-Lik: -10354.353, Max-Change: 0.00103
Iteration: 22, Log-Lik: -10354.342, Max-Change: 0.00088
Iteration: 23, Log-Lik: -10354.334, Max-Change: 0.00074
Iteration: 24, Log-Lik: -10354.329, Max-Change: 0.00062
Iteration: 25, Log-Lik: -10354.325, Max-Change: 0.00053
Iteration: 26, Log-Lik: -10354.322, Max-Change: 0.00045
Iteration: 27, Log-Lik: -10354.320, Max-Change: 0.00038
Iteration: 28, Log-Lik: -10354.318, Max-Change: 0.00032
Iteration: 29, Log-Lik: -10354.317, Max-Change: 0.00027
Iteration: 30, Log-Lik: -10354.316, Max-Change: 0.00023
Iteration: 31, Log-Lik: -10354.316, Max-Change: 0.00020
Iteration: 32, Log-Lik: -10354.315, Max-Change: 0.00017
Iteration: 33, Log-Lik: -10354.315, Max-Change: 0.00014
Iteration: 34, Log-Lik: -10354.315, Max-Change: 0.00012
Iteration: 35, Log-Lik: -10354.315, Max-Change: 0.00010
Iteration: 36, Log-Lik: -10354.315, Max-Change: 0.00009
anova(mod1b, mod1)
#>            AIC    SABIC       HQ      BIC    logLik    X2 df     p
#> mod1b 20754.63 20794.46 20797.53 20867.51 -10354.32               
#> mod1  20756.61 20798.17 20801.38 20874.40 -10354.31 0.018  1 0.893

# compare to mixedmirt() version of the same model
mod1.mixed <- mixedmirt(dat, 1, itemtype='Rasch',
                        covdata=covdata, lr.fixed = ~ X1 + X2 + X3, SE=TRUE)
#> 
Stage 1 = 1, CDLL = -13584.5, AR(0.80) = [0.48], Max-Change = 0.1317
Stage 1 = 2, CDLL = -13479.0, AR(0.80) = [0.50], Max-Change = 0.1109
Stage 1 = 3, CDLL = -13421.6, AR(0.80) = [0.49], Max-Change = 0.0928
Stage 1 = 4, CDLL = -13328.9, AR(0.80) = [0.49], Max-Change = 0.2000
Stage 1 = 5, CDLL = -13198.2, AR(0.80) = [0.45], Max-Change = 0.0832
Stage 1 = 6, CDLL = -13154.8, AR(0.80) = [0.45], Max-Change = 0.0494
Stage 1 = 7, CDLL = -13096.7, AR(0.80) = [0.44], Max-Change = 0.0447
Stage 1 = 8, CDLL = -13067.2, AR(0.80) = [0.48], Max-Change = 0.0467
Stage 1 = 9, CDLL = -13011.7, AR(0.80) = [0.46], Max-Change = 0.0290
Stage 1 = 10, CDLL = -12943.9, AR(0.80) = [0.45], Max-Change = 0.0373
Stage 1 = 11, CDLL = -12900.4, AR(0.80) = [0.44], Max-Change = 0.0231
Stage 1 = 12, CDLL = -12899.5, AR(0.80) = [0.41], Max-Change = 0.0177
Stage 1 = 13, CDLL = -12885.2, AR(0.80) = [0.44], Max-Change = 0.0250
Stage 1 = 14, CDLL = -12834.1, AR(0.80) = [0.42], Max-Change = 0.0135
Stage 1 = 15, CDLL = -12815.7, AR(0.80) = [0.41], Max-Change = 0.0126
Stage 1 = 16, CDLL = -12785.4, AR(0.80) = [0.40], Max-Change = 0.0102
Stage 1 = 17, CDLL = -12768.8, AR(0.80) = [0.40], Max-Change = 0.0078
Stage 1 = 18, CDLL = -12808.1, AR(0.80) = [0.43], Max-Change = 0.0107
Stage 1 = 19, CDLL = -12769.0, AR(0.80) = [0.41], Max-Change = 0.0102
Stage 1 = 20, CDLL = -12766.7, AR(0.80) = [0.41], Max-Change = 0.0073
Stage 1 = 21, CDLL = -12764.9, AR(0.80) = [0.39], Max-Change = 0.0075
Stage 1 = 22, CDLL = -12727.5, AR(0.80) = [0.39], Max-Change = 0.0059
Stage 1 = 23, CDLL = -12703.4, AR(0.80) = [0.41], Max-Change = 0.0071
Stage 1 = 24, CDLL = -12693.6, AR(0.80) = [0.38], Max-Change = 0.0106
Stage 1 = 25, CDLL = -12646.9, AR(0.80) = [0.39], Max-Change = 0.0133
Stage 1 = 26, CDLL = -12654.6, AR(0.80) = [0.39], Max-Change = 0.0034
Stage 1 = 27, CDLL = -12707.2, AR(0.80) = [0.40], Max-Change = 0.0015
Stage 1 = 28, CDLL = -12677.6, AR(0.80) = [0.39], Max-Change = 0.0054
Stage 1 = 29, CDLL = -12647.9, AR(0.80) = [0.37], Max-Change = 0.0074
Stage 1 = 30, CDLL = -12648.9, AR(0.80) = [0.41], Max-Change = 0.0034
Stage 1 = 31, CDLL = -12657.8, AR(0.80) = [0.40], Max-Change = 0.0037
Stage 1 = 32, CDLL = -12632.9, AR(0.80) = [0.39], Max-Change = 0.0062
Stage 1 = 33, CDLL = -12602.2, AR(0.80) = [0.35], Max-Change = 0.0042
Stage 1 = 34, CDLL = -12619.3, AR(0.80) = [0.36], Max-Change = 0.0055
Stage 1 = 35, CDLL = -12606.9, AR(0.80) = [0.38], Max-Change = 0.0040
Stage 1 = 36, CDLL = -12626.2, AR(0.80) = [0.38], Max-Change = 0.0024
Stage 1 = 37, CDLL = -12611.9, AR(0.80) = [0.40], Max-Change = 0.0034
Stage 1 = 38, CDLL = -12544.1, AR(0.80) = [0.38], Max-Change = 0.0049
Stage 1 = 39, CDLL = -12588.6, AR(0.80) = [0.36], Max-Change = 0.0043
Stage 1 = 40, CDLL = -12586.7, AR(0.80) = [0.35], Max-Change = 0.0024
Stage 1 = 41, CDLL = -12641.1, AR(0.80) = [0.38], Max-Change = 0.0024
Stage 1 = 42, CDLL = -12571.9, AR(0.80) = [0.36], Max-Change = 0.0032
Stage 1 = 43, CDLL = -12611.6, AR(0.80) = [0.35], Max-Change = 0.0032
Stage 1 = 44, CDLL = -12564.6, AR(0.80) = [0.37], Max-Change = 0.0040
Stage 1 = 45, CDLL = -12533.6, AR(0.80) = [0.36], Max-Change = 0.0022
Stage 1 = 46, CDLL = -12581.0, AR(0.80) = [0.36], Max-Change = 0.0031
Stage 1 = 47, CDLL = -12571.9, AR(0.80) = [0.40], Max-Change = 0.0010
Stage 1 = 48, CDLL = -12549.5, AR(0.80) = [0.35], Max-Change = 0.0035
Stage 1 = 49, CDLL = -12594.2, AR(0.80) = [0.35], Max-Change = 0.0025
Stage 1 = 50, CDLL = -12568.8, AR(0.80) = [0.37], Max-Change = 0.0016
Stage 1 = 51, CDLL = -12559.1, AR(0.80) = [0.37], Max-Change = 0.0031
Stage 1 = 52, CDLL = -12553.3, AR(0.80) = [0.35], Max-Change = 0.0033
Stage 1 = 53, CDLL = -12546.3, AR(0.80) = [0.33], Max-Change = 0.0033
Stage 1 = 54, CDLL = -12538.3, AR(0.80) = [0.38], Max-Change = 0.0024
Stage 1 = 55, CDLL = -12507.0, AR(0.80) = [0.37], Max-Change = 0.0025
Stage 1 = 56, CDLL = -12527.1, AR(0.80) = [0.36], Max-Change = 0.0029
Stage 1 = 57, CDLL = -12571.3, AR(0.80) = [0.34], Max-Change = 0.0014
Stage 1 = 58, CDLL = -12527.8, AR(0.80) = [0.37], Max-Change = 0.0025
Stage 1 = 59, CDLL = -12547.8, AR(0.80) = [0.34], Max-Change = 0.0028
Stage 1 = 60, CDLL = -12528.4, AR(0.80) = [0.35], Max-Change = 0.0024
Stage 1 = 61, CDLL = -12542.2, AR(0.80) = [0.34], Max-Change = 0.0023
Stage 1 = 62, CDLL = -12478.8, AR(0.80) = [0.32], Max-Change = 0.0028
Stage 1 = 63, CDLL = -12517.5, AR(0.80) = [0.35], Max-Change = 0.0025
Stage 1 = 64, CDLL = -12532.8, AR(0.80) = [0.35], Max-Change = 0.0033
Stage 1 = 65, CDLL = -12504.5, AR(0.80) = [0.35], Max-Change = 0.0055
Stage 1 = 66, CDLL = -12499.1, AR(0.80) = [0.35], Max-Change = 0.0030
Stage 1 = 67, CDLL = -12487.2, AR(0.80) = [0.34], Max-Change = 0.0016
Stage 1 = 68, CDLL = -12507.0, AR(0.80) = [0.35], Max-Change = 0.0034
Stage 1 = 69, CDLL = -12469.7, AR(0.80) = [0.36], Max-Change = 0.0030
Stage 1 = 70, CDLL = -12469.5, AR(0.80) = [0.33], Max-Change = 0.0026
Stage 1 = 71, CDLL = -12494.0, AR(0.80) = [0.32], Max-Change = 0.0033
Stage 1 = 72, CDLL = -12483.8, AR(0.80) = [0.34], Max-Change = 0.0006
Stage 1 = 73, CDLL = -12530.8, AR(0.80) = [0.34], Max-Change = 0.0026
Stage 1 = 74, CDLL = -12453.8, AR(0.80) = [0.35], Max-Change = 0.0041
Stage 1 = 75, CDLL = -12472.5, AR(0.80) = [0.33], Max-Change = 0.0029
Stage 1 = 76, CDLL = -12472.4, AR(0.80) = [0.35], Max-Change = 0.0015
Stage 1 = 77, CDLL = -12497.3, AR(0.80) = [0.33], Max-Change = 0.0026
Stage 1 = 78, CDLL = -12491.2, AR(0.80) = [0.33], Max-Change = 0.0024
Stage 1 = 79, CDLL = -12494.9, AR(0.80) = [0.37], Max-Change = 0.0050
Stage 1 = 80, CDLL = -12505.2, AR(0.80) = [0.35], Max-Change = 0.0037
Stage 1 = 81, CDLL = -12494.2, AR(0.80) = [0.34], Max-Change = 0.0017
Stage 1 = 82, CDLL = -12448.8, AR(0.80) = [0.33], Max-Change = 0.0027
Stage 1 = 83, CDLL = -12461.9, AR(0.80) = [0.33], Max-Change = 0.0042
Stage 1 = 84, CDLL = -12473.8, AR(0.80) = [0.33], Max-Change = 0.0027
Stage 1 = 85, CDLL = -12510.0, AR(0.80) = [0.33], Max-Change = 0.0037
Stage 1 = 86, CDLL = -12447.5, AR(0.80) = [0.33], Max-Change = 0.0048
Stage 1 = 87, CDLL = -12514.8, AR(0.80) = [0.34], Max-Change = 0.0020
Stage 1 = 88, CDLL = -12468.7, AR(0.80) = [0.33], Max-Change = 0.0024
Stage 1 = 89, CDLL = -12473.9, AR(0.80) = [0.36], Max-Change = 0.0039
Stage 1 = 90, CDLL = -12470.2, AR(0.80) = [0.34], Max-Change = 0.0027
Stage 1 = 91, CDLL = -12479.5, AR(0.80) = [0.34], Max-Change = 0.0018
Stage 1 = 92, CDLL = -12506.0, AR(0.80) = [0.36], Max-Change = 0.0015
Stage 1 = 93, CDLL = -12470.9, AR(0.80) = [0.33], Max-Change = 0.0019
Stage 1 = 94, CDLL = -12479.8, AR(0.80) = [0.34], Max-Change = 0.0022
Stage 1 = 95, CDLL = -12493.4, AR(0.80) = [0.34], Max-Change = 0.0019
Stage 1 = 96, CDLL = -12451.0, AR(0.80) = [0.35], Max-Change = 0.0022
Stage 1 = 97, CDLL = -12469.9, AR(0.80) = [0.32], Max-Change = 0.0024
Stage 1 = 98, CDLL = -12454.4, AR(0.80) = [0.32], Max-Change = 0.0024
Stage 1 = 99, CDLL = -12431.6, AR(0.80) = [0.32], Max-Change = 0.0018
Stage 1 = 100, CDLL = -12465.6, AR(0.80) = [0.33], Max-Change = 0.0029
Stage 1 = 101, CDLL = -12462.7, AR(0.80) = [0.34], Max-Change = 0.0042
Stage 1 = 102, CDLL = -12474.7, AR(0.80) = [0.37], Max-Change = 0.0017
Stage 1 = 103, CDLL = -12486.8, AR(0.80) = [0.36], Max-Change = 0.0013
Stage 1 = 104, CDLL = -12480.0, AR(0.80) = [0.33], Max-Change = 0.0029
Stage 1 = 105, CDLL = -12483.6, AR(0.80) = [0.32], Max-Change = 0.0016
Stage 1 = 106, CDLL = -12454.9, AR(0.80) = [0.34], Max-Change = 0.0027
Stage 1 = 107, CDLL = -12459.3, AR(0.80) = [0.36], Max-Change = 0.0021
Stage 1 = 108, CDLL = -12471.6, AR(0.80) = [0.34], Max-Change = 0.0027
Stage 1 = 109, CDLL = -12434.3, AR(0.80) = [0.34], Max-Change = 0.0026
Stage 1 = 110, CDLL = -12468.7, AR(0.80) = [0.30], Max-Change = 0.0019
Stage 1 = 111, CDLL = -12433.6, AR(0.80) = [0.31], Max-Change = 0.0008
Stage 1 = 112, CDLL = -12433.0, AR(0.80) = [0.34], Max-Change = 0.0019
Stage 1 = 113, CDLL = -12395.6, AR(0.80) = [0.33], Max-Change = 0.0043
Stage 1 = 114, CDLL = -12448.0, AR(0.80) = [0.32], Max-Change = 0.0016
Stage 1 = 115, CDLL = -12465.5, AR(0.80) = [0.30], Max-Change = 0.0032
Stage 1 = 116, CDLL = -12427.6, AR(0.80) = [0.34], Max-Change = 0.0024
Stage 1 = 117, CDLL = -12479.9, AR(0.80) = [0.33], Max-Change = 0.0043
Stage 1 = 118, CDLL = -12400.1, AR(0.80) = [0.33], Max-Change = 0.0022
Stage 1 = 119, CDLL = -12440.6, AR(0.80) = [0.33], Max-Change = 0.0026
Stage 1 = 120, CDLL = -12481.2, AR(0.80) = [0.33], Max-Change = 0.0037
Stage 1 = 121, CDLL = -12517.2, AR(0.80) = [0.34], Max-Change = 0.0034
Stage 1 = 122, CDLL = -12436.1, AR(0.80) = [0.34], Max-Change = 0.0029
Stage 1 = 123, CDLL = -12442.9, AR(0.80) = [0.32], Max-Change = 0.0016
Stage 1 = 124, CDLL = -12432.6, AR(0.80) = [0.33], Max-Change = 0.0033
Stage 1 = 125, CDLL = -12463.8, AR(0.80) = [0.35], Max-Change = 0.0014
Stage 1 = 126, CDLL = -12444.8, AR(0.80) = [0.35], Max-Change = 0.0035
Stage 1 = 127, CDLL = -12426.3, AR(0.80) = [0.34], Max-Change = 0.0023
Stage 1 = 128, CDLL = -12422.5, AR(0.80) = [0.33], Max-Change = 0.0027
Stage 1 = 129, CDLL = -12427.6, AR(0.80) = [0.34], Max-Change = 0.0010
Stage 1 = 130, CDLL = -12428.1, AR(0.80) = [0.32], Max-Change = 0.0018
Stage 1 = 131, CDLL = -12444.4, AR(0.80) = [0.34], Max-Change = 0.0015
Stage 1 = 132, CDLL = -12419.9, AR(0.80) = [0.32], Max-Change = 0.0009
Stage 1 = 133, CDLL = -12405.0, AR(0.80) = [0.35], Max-Change = 0.0017
Stage 1 = 134, CDLL = -12413.5, AR(0.80) = [0.32], Max-Change = 0.0026
Stage 1 = 135, CDLL = -12447.8, AR(0.80) = [0.33], Max-Change = 0.0035
Stage 1 = 136, CDLL = -12457.1, AR(0.80) = [0.30], Max-Change = 0.0030
Stage 1 = 137, CDLL = -12419.8, AR(0.80) = [0.31], Max-Change = 0.0052
Stage 1 = 138, CDLL = -12422.8, AR(0.80) = [0.32], Max-Change = 0.0011
Stage 1 = 139, CDLL = -12424.7, AR(0.80) = [0.31], Max-Change = 0.0025
Stage 1 = 140, CDLL = -12407.2, AR(0.80) = [0.31], Max-Change = 0.0043
Stage 1 = 141, CDLL = -12410.0, AR(0.80) = [0.32], Max-Change = 0.0016
Stage 1 = 142, CDLL = -12461.0, AR(0.80) = [0.35], Max-Change = 0.0028
Stage 1 = 143, CDLL = -12397.1, AR(0.80) = [0.31], Max-Change = 0.0021
Stage 1 = 144, CDLL = -12418.7, AR(0.80) = [0.32], Max-Change = 0.0028
Stage 1 = 145, CDLL = -12411.9, AR(0.80) = [0.30], Max-Change = 0.0026
Stage 1 = 146, CDLL = -12416.9, AR(0.80) = [0.30], Max-Change = 0.0032
Stage 1 = 147, CDLL = -12398.5, AR(0.80) = [0.32], Max-Change = 0.0020
Stage 1 = 148, CDLL = -12440.1, AR(0.80) = [0.32], Max-Change = 0.0021
Stage 1 = 149, CDLL = -12483.3, AR(0.80) = [0.34], Max-Change = 0.0051
Stage 1 = 150, CDLL = -12424.5, AR(0.48) = [0.41], Max-Change = 0.0025
Stage 2 = 1, CDLL = -12462.7, AR(0.48) = [0.39], Max-Change = 0.0024
Stage 2 = 2, CDLL = -12414.2, AR(0.48) = [0.41], Max-Change = 0.0018
Stage 2 = 3, CDLL = -12431.2, AR(0.48) = [0.40], Max-Change = 0.0014
Stage 2 = 4, CDLL = -12442.6, AR(0.48) = [0.41], Max-Change = 0.0029
Stage 2 = 5, CDLL = -12444.9, AR(0.48) = [0.40], Max-Change = 0.0029
Stage 2 = 6, CDLL = -12436.7, AR(0.48) = [0.36], Max-Change = 0.0009
Stage 2 = 7, CDLL = -12444.0, AR(0.48) = [0.39], Max-Change = 0.0022
Stage 2 = 8, CDLL = -12439.1, AR(0.48) = [0.39], Max-Change = 0.0029
Stage 2 = 9, CDLL = -12441.3, AR(0.48) = [0.38], Max-Change = 0.0015
Stage 2 = 10, CDLL = -12428.5, AR(0.48) = [0.41], Max-Change = 0.0018
Stage 2 = 11, CDLL = -12404.3, AR(0.48) = [0.40], Max-Change = 0.0032
Stage 2 = 12, CDLL = -12413.5, AR(0.48) = [0.39], Max-Change = 0.0027
Stage 2 = 13, CDLL = -12407.0, AR(0.48) = [0.38], Max-Change = 0.0011
Stage 2 = 14, CDLL = -12414.1, AR(0.48) = [0.39], Max-Change = 0.0019
Stage 2 = 15, CDLL = -12432.6, AR(0.48) = [0.41], Max-Change = 0.0027
Stage 2 = 16, CDLL = -12429.4, AR(0.48) = [0.40], Max-Change = 0.0039
Stage 2 = 17, CDLL = -12442.9, AR(0.48) = [0.38], Max-Change = 0.0032
Stage 2 = 18, CDLL = -12437.8, AR(0.48) = [0.40], Max-Change = 0.0031
Stage 2 = 19, CDLL = -12380.4, AR(0.48) = [0.39], Max-Change = 0.0026
Stage 2 = 20, CDLL = -12441.2, AR(0.48) = [0.39], Max-Change = 0.0036
Stage 2 = 21, CDLL = -12445.9, AR(0.48) = [0.41], Max-Change = 0.0021
Stage 2 = 22, CDLL = -12446.7, AR(0.48) = [0.38], Max-Change = 0.0019
Stage 2 = 23, CDLL = -12468.6, AR(0.48) = [0.39], Max-Change = 0.0011
Stage 2 = 24, CDLL = -12412.5, AR(0.48) = [0.37], Max-Change = 0.0029
Stage 2 = 25, CDLL = -12423.6, AR(0.48) = [0.41], Max-Change = 0.0017
Stage 2 = 26, CDLL = -12445.9, AR(0.48) = [0.38], Max-Change = 0.0012
Stage 2 = 27, CDLL = -12422.2, AR(0.48) = [0.40], Max-Change = 0.0014
Stage 2 = 28, CDLL = -12404.3, AR(0.48) = [0.38], Max-Change = 0.0040
Stage 2 = 29, CDLL = -12428.6, AR(0.48) = [0.39], Max-Change = 0.0017
Stage 2 = 30, CDLL = -12377.5, AR(0.48) = [0.42], Max-Change = 0.0028
Stage 2 = 31, CDLL = -12399.6, AR(0.48) = [0.39], Max-Change = 0.0019
Stage 2 = 32, CDLL = -12377.5, AR(0.48) = [0.40], Max-Change = 0.0015
Stage 2 = 33, CDLL = -12443.2, AR(0.48) = [0.38], Max-Change = 0.0026
Stage 2 = 34, CDLL = -12399.4, AR(0.48) = [0.37], Max-Change = 0.0026
Stage 2 = 35, CDLL = -12412.2, AR(0.48) = [0.38], Max-Change = 0.0025
Stage 2 = 36, CDLL = -12433.0, AR(0.48) = [0.37], Max-Change = 0.0032
Stage 2 = 37, CDLL = -12421.9, AR(0.48) = [0.39], Max-Change = 0.0019
Stage 2 = 38, CDLL = -12429.6, AR(0.48) = [0.41], Max-Change = 0.0019
Stage 2 = 39, CDLL = -12432.6, AR(0.48) = [0.42], Max-Change = 0.0025
Stage 2 = 40, CDLL = -12439.7, AR(0.48) = [0.39], Max-Change = 0.0020
Stage 2 = 41, CDLL = -12402.9, AR(0.48) = [0.41], Max-Change = 0.0027
Stage 2 = 42, CDLL = -12416.4, AR(0.48) = [0.40], Max-Change = 0.0019
Stage 2 = 43, CDLL = -12422.7, AR(0.48) = [0.40], Max-Change = 0.0011
Stage 2 = 44, CDLL = -12430.5, AR(0.48) = [0.39], Max-Change = 0.0034
Stage 2 = 45, CDLL = -12420.2, AR(0.48) = [0.40], Max-Change = 0.0035
Stage 2 = 46, CDLL = -12421.3, AR(0.48) = [0.38], Max-Change = 0.0029
Stage 2 = 47, CDLL = -12451.3, AR(0.48) = [0.38], Max-Change = 0.0015
Stage 2 = 48, CDLL = -12407.2, AR(0.48) = [0.38], Max-Change = 0.0034
Stage 2 = 49, CDLL = -12450.2, AR(0.48) = [0.41], Max-Change = 0.0019
Stage 2 = 50, CDLL = -12436.2, AR(0.48) = [0.41], Max-Change = 0.0015
Stage 2 = 51, CDLL = -12466.7, AR(0.48) = [0.39], Max-Change = 0.0013
Stage 2 = 52, CDLL = -12431.8, AR(0.48) = [0.42], Max-Change = 0.0031
Stage 2 = 53, CDLL = -12472.8, AR(0.48) = [0.41], Max-Change = 0.0028
Stage 2 = 54, CDLL = -12436.0, AR(0.48) = [0.38], Max-Change = 0.0017
Stage 2 = 55, CDLL = -12460.0, AR(0.48) = [0.38], Max-Change = 0.0023
Stage 2 = 56, CDLL = -12474.9, AR(0.48) = [0.39], Max-Change = 0.0033
Stage 2 = 57, CDLL = -12469.2, AR(0.48) = [0.40], Max-Change = 0.0031
Stage 2 = 58, CDLL = -12460.7, AR(0.48) = [0.42], Max-Change = 0.0019
Stage 2 = 59, CDLL = -12459.3, AR(0.48) = [0.39], Max-Change = 0.0034
Stage 2 = 60, CDLL = -12435.4, AR(0.48) = [0.39], Max-Change = 0.0025
Stage 2 = 61, CDLL = -12469.0, AR(0.48) = [0.44], Max-Change = 0.0028
Stage 2 = 62, CDLL = -12441.3, AR(0.48) = [0.40], Max-Change = 0.0020
Stage 2 = 63, CDLL = -12461.0, AR(0.48) = [0.40], Max-Change = 0.0027
Stage 2 = 64, CDLL = -12475.8, AR(0.48) = [0.41], Max-Change = 0.0023
Stage 2 = 65, CDLL = -12425.7, AR(0.48) = [0.44], Max-Change = 0.0053
Stage 2 = 66, CDLL = -12432.4, AR(0.48) = [0.40], Max-Change = 0.0036
Stage 2 = 67, CDLL = -12399.3, AR(0.48) = [0.41], Max-Change = 0.0026
Stage 2 = 68, CDLL = -12441.5, AR(0.48) = [0.40], Max-Change = 0.0024
Stage 2 = 69, CDLL = -12429.1, AR(0.48) = [0.43], Max-Change = 0.0024
Stage 2 = 70, CDLL = -12418.9, AR(0.48) = [0.41], Max-Change = 0.0029
Stage 2 = 71, CDLL = -12412.9, AR(0.48) = [0.38], Max-Change = 0.0021
Stage 2 = 72, CDLL = -12398.5, AR(0.48) = [0.42], Max-Change = 0.0011
Stage 2 = 73, CDLL = -12435.3, AR(0.48) = [0.40], Max-Change = 0.0020
Stage 2 = 74, CDLL = -12457.0, AR(0.48) = [0.41], Max-Change = 0.0018
Stage 2 = 75, CDLL = -12448.7, AR(0.48) = [0.40], Max-Change = 0.0020
Stage 2 = 76, CDLL = -12444.0, AR(0.48) = [0.36], Max-Change = 0.0016
Stage 2 = 77, CDLL = -12405.8, AR(0.48) = [0.41], Max-Change = 0.0012
Stage 2 = 78, CDLL = -12441.2, AR(0.48) = [0.42], Max-Change = 0.0025
Stage 2 = 79, CDLL = -12432.2, AR(0.48) = [0.39], Max-Change = 0.0040
Stage 2 = 80, CDLL = -12434.3, AR(0.48) = [0.39], Max-Change = 0.0030
Stage 2 = 81, CDLL = -12443.5, AR(0.48) = [0.38], Max-Change = 0.0019
Stage 2 = 82, CDLL = -12431.2, AR(0.48) = [0.39], Max-Change = 0.0039
Stage 2 = 83, CDLL = -12436.8, AR(0.48) = [0.42], Max-Change = 0.0017
Stage 2 = 84, CDLL = -12407.6, AR(0.48) = [0.41], Max-Change = 0.0022
Stage 2 = 85, CDLL = -12401.1, AR(0.48) = [0.41], Max-Change = 0.0013
Stage 2 = 86, CDLL = -12423.1, AR(0.48) = [0.41], Max-Change = 0.0013
Stage 2 = 87, CDLL = -12412.9, AR(0.48) = [0.36], Max-Change = 0.0034
Stage 2 = 88, CDLL = -12408.7, AR(0.48) = [0.38], Max-Change = 0.0010
Stage 2 = 89, CDLL = -12409.0, AR(0.48) = [0.37], Max-Change = 0.0014
Stage 2 = 90, CDLL = -12388.8, AR(0.48) = [0.37], Max-Change = 0.0015
Stage 2 = 91, CDLL = -12396.3, AR(0.48) = [0.39], Max-Change = 0.0014
Stage 2 = 92, CDLL = -12428.9, AR(0.48) = [0.39], Max-Change = 0.0011
Stage 2 = 93, CDLL = -12438.6, AR(0.48) = [0.43], Max-Change = 0.0036
Stage 2 = 94, CDLL = -12446.3, AR(0.48) = [0.39], Max-Change = 0.0013
Stage 2 = 95, CDLL = -12445.6, AR(0.48) = [0.41], Max-Change = 0.0020
Stage 2 = 96, CDLL = -12394.4, AR(0.48) = [0.40], Max-Change = 0.0024
Stage 2 = 97, CDLL = -12416.8, AR(0.48) = [0.41], Max-Change = 0.0018
Stage 2 = 98, CDLL = -12438.3, AR(0.48) = [0.38], Max-Change = 0.0029
Stage 2 = 99, CDLL = -12434.5, AR(0.48) = [0.41], Max-Change = 0.0051
Stage 2 = 100, CDLL = -12388.0, AR(0.48) = [0.41], Max-Change = 0.0023
Stage 3 = 1, CDLL = -12427.4, AR(0.48) = [0.40], gam = 0.0000, Max-Change = 0.0000
Stage 3 = 2, CDLL = -12439.3, AR(0.48) = [0.41], gam = 0.1778, Max-Change = 0.0033
Stage 3 = 3, CDLL = -12432.6, AR(0.48) = [0.40], gam = 0.1057, Max-Change = 0.0012
Stage 3 = 4, CDLL = -12400.5, AR(0.48) = [0.41], gam = 0.0780, Max-Change = 0.0010
Stage 3 = 5, CDLL = -12451.1, AR(0.48) = [0.41], gam = 0.0629, Max-Change = 0.0005
Stage 3 = 6, CDLL = -12453.3, AR(0.48) = [0.41], gam = 0.0532, Max-Change = 0.0008
Stage 3 = 7, CDLL = -12441.9, AR(0.48) = [0.40], gam = 0.0464, Max-Change = 0.0006

#> 
#> Calculating information matrix...
#> 
#> Calculating log-likelihood...
coef(mod1.mixed)
#> $Item_1
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_2
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_3
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_4
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_5
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_6
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_7
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_8
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_9
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_10
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_11
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_12
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_13
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_14
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_15
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_16
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_17
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_18
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_19
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $Item_20
#>         (Intercept) a1  d  g  u
#> par          -0.131  1  0  0  1
#> CI_2.5       -0.166 NA NA NA NA
#> CI_97.5      -0.097 NA NA NA NA
#> 
#> $GroupPars
#>         MEAN_1 COV_11
#> par          0  0.087
#> CI_2.5      NA  0.068
#> CI_97.5     NA  0.105
#> 
#> $lr.betas
#>         F1_(Intercept) F1_X1  F1_X2  F1_X3
#> par                  0 0.409 -0.795 -0.007
#> CI_2.5              NA 0.376 -0.837 -0.040
#> CI_97.5             NA 0.441 -0.753  0.027
#> 
coef(mod1.mixed, printSE=TRUE)
#> $Item_1
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_2
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_3
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_4
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_5
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_6
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_7
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_8
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_9
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_10
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_11
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_12
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_13
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_14
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_15
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_16
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_17
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_18
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_19
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $Item_20
#>     (Intercept) a1  d    g   u
#> par      -0.131  1  0 -999 999
#> SE        0.018 NA NA   NA  NA
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0  0.087
#> SE      NA  0.009
#> 
#> $lr.betas
#>     F1_(Intercept) F1_X1  F1_X2  F1_X3
#> par              0 0.409 -0.795 -0.007
#> SE              NA 0.016  0.021  0.017
#> 

# draw plausible values for secondary analyses
pv <- fscores(mod1, plausible.draws = 10)
pvmods <- lapply(pv, function(x, covdata) lm(x ~ covdata$X1 + covdata$X2),
                 covdata=covdata)
# population characteristics recovered well, and can be averaged over
so <- lapply(pvmods, summary)
so
#> [[1]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.58018 -0.29446  0.00306  0.28514  1.80285 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  0.001826   0.014870   0.123    0.902    
#> covdata$X1   0.522138   0.014934  34.963   <2e-16 ***
#> covdata$X2  -1.007205   0.015180 -66.353   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.47 on 997 degrees of freedom
#> Multiple R-squared:  0.8436,	Adjusted R-squared:  0.8433 
#> F-statistic:  2690 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[2]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.41425 -0.31655  0.01905  0.30417  1.32953 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.007369   0.014386  -0.512    0.609    
#> covdata$X1   0.501445   0.014447  34.708   <2e-16 ***
#> covdata$X2  -1.012151   0.014685 -68.924   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4547 on 997 degrees of freedom
#> Multiple R-squared:  0.8512,	Adjusted R-squared:  0.8509 
#> F-statistic:  2851 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[3]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.40267 -0.31055  0.04396  0.30822  1.36044 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.003617   0.014556  -0.248    0.804    
#> covdata$X1   0.502951   0.014618  34.406   <2e-16 ***
#> covdata$X2  -0.999167   0.014859 -67.245   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4601 on 997 degrees of freedom
#> Multiple R-squared:  0.8456,	Adjusted R-squared:  0.8453 
#> F-statistic:  2731 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[4]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.37771 -0.32496  0.00811  0.32851  1.59946 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  0.01587    0.01497    1.06    0.289    
#> covdata$X1   0.52893    0.01504   35.18   <2e-16 ***
#> covdata$X2  -0.99322    0.01528  -64.98   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4732 on 997 degrees of freedom
#> Multiple R-squared:  0.8396,	Adjusted R-squared:  0.8393 
#> F-statistic:  2609 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[5]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.57210 -0.30913 -0.00413  0.31196  1.33557 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  0.006511   0.014338   0.454     0.65    
#> covdata$X1   0.511708   0.014400  35.536   <2e-16 ***
#> covdata$X2  -1.006448   0.014637 -68.763   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4532 on 997 degrees of freedom
#> Multiple R-squared:  0.8519,	Adjusted R-squared:  0.8516 
#> F-statistic:  2866 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[6]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.66029 -0.32641 -0.00233  0.31263  1.67948 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept)  0.006891   0.014906   0.462    0.644    
#> covdata$X1   0.504434   0.014970  33.696   <2e-16 ***
#> covdata$X2  -0.988252   0.015216 -64.947   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4711 on 997 degrees of freedom
#> Multiple R-squared:  0.8371,	Adjusted R-squared:  0.8367 
#> F-statistic:  2561 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[7]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.56408 -0.31059 -0.00792  0.31611  1.60839 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.01777    0.01461  -1.216    0.224    
#> covdata$X1   0.52230    0.01467  35.596   <2e-16 ***
#> covdata$X2  -1.01685    0.01491 -68.180   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4618 on 997 degrees of freedom
#> Multiple R-squared:  0.8502,	Adjusted R-squared:  0.8499 
#> F-statistic:  2829 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[8]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.70215 -0.28200  0.00583  0.29951  1.32691 
#> 
#> Coefficients:
#>              Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.002541   0.014172  -0.179    0.858    
#> covdata$X1   0.515759   0.014233  36.237   <2e-16 ***
#> covdata$X2  -0.985688   0.014467 -68.133   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4479 on 997 degrees of freedom
#> Multiple R-squared:  0.851,	Adjusted R-squared:  0.8507 
#> F-statistic:  2847 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[9]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.35302 -0.31098 -0.00154  0.31738  1.50051 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.02344    0.01448  -1.619    0.106    
#> covdata$X1   0.53585    0.01454  36.848   <2e-16 ***
#> covdata$X2  -0.99557    0.01478 -67.352   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4577 on 997 degrees of freedom
#> Multiple R-squared:  0.8496,	Adjusted R-squared:  0.8493 
#> F-statistic:  2816 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 
#> [[10]]
#> 
#> Call:
#> lm(formula = x ~ covdata$X1 + covdata$X2)
#> 
#> Residuals:
#>      Min       1Q   Median       3Q      Max 
#> -1.64819 -0.32132 -0.01963  0.32821  1.55474 
#> 
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)    
#> (Intercept) -0.01220    0.01483  -0.823    0.411    
#> covdata$X1   0.51641    0.01489  34.671   <2e-16 ***
#> covdata$X2  -1.01707    0.01514 -67.179   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.4688 on 997 degrees of freedom
#> Multiple R-squared:  0.8458,	Adjusted R-squared:  0.8455 
#> F-statistic:  2734 on 2 and 997 DF,  p-value: < 2.2e-16
#> 
#> 

# compute Rubin's multiple imputation average
par <- lapply(so, function(x) x$coefficients[, 'Estimate'])
SEpar <- lapply(so, function(x) x$coefficients[, 'Std. Error'])
averageMI(par, SEpar)
#>                par SEpar       t     df     p
#> (Intercept) -0.004 0.019  -0.186 49.256 0.213
#> covdata$X1   0.516 0.019  27.255 56.399     0
#> covdata$X2  -1.002 0.019 -52.338 58.281     0

############
# Example using Gauss-Hermite quadrature with custom input functions

if (FALSE) { # \dontrun{
library(fastGHQuad)
data(SAT12)
data <- key2binary(SAT12,
                   key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
GH <- gaussHermiteData(50)
Theta <- matrix(GH$x)

# This prior works for uni- and multi-dimensional models
prior <- function(Theta, Etable){
    P <- grid <- GH$w / sqrt(pi)
    if(ncol(Theta) > 1)
        for(i in 2:ncol(Theta))
            P <- expand.grid(P, grid)
     if(!is.vector(P)) P <- apply(P, 1, prod)
     P
}

GHmod1 <- mirt(data, 1, optimizer = 'NR',
              technical = list(customTheta = Theta, customPriorFun = prior))
coef(GHmod1, simplify=TRUE)

Theta2 <- as.matrix(expand.grid(Theta, Theta))
GHmod2 <- mirt(data, 2, optimizer = 'NR', TOL = .0002,
              technical = list(customTheta = Theta2, customPriorFun = prior))
summary(GHmod2, suppress=.2)
} # }

############
# Davidian curve example

dat <- key2binary(SAT12,
                   key = c(1,4,5,2,3,1,2,1,3,1,2,4,2,1,5,3,4,4,1,4,3,3,4,1,3,5,1,3,1,5,4,5))
dav <- mirt(dat, 1, dentype = 'Davidian-4') # use four smoothing parameters
#> 
Iteration: 1, Log-Lik: -9613.786, Max-Change: 1.59379
Iteration: 2, Log-Lik: -9462.908, Max-Change: 0.38466
Iteration: 3, Log-Lik: -9444.402, Max-Change: 0.23488
Iteration: 4, Log-Lik: -9438.567, Max-Change: 0.16574
Iteration: 5, Log-Lik: -9435.700, Max-Change: 0.12922
Iteration: 6, Log-Lik: -9433.789, Max-Change: 0.10223
Iteration: 7, Log-Lik: -9432.154, Max-Change: 0.04878
Iteration: 8, Log-Lik: -9432.104, Max-Change: 0.03598
Iteration: 9, Log-Lik: -9432.230, Max-Change: 0.03555
Iteration: 10, Log-Lik: -9432.055, Max-Change: 0.02082
Iteration: 11, Log-Lik: -9432.099, Max-Change: 0.00970
Iteration: 12, Log-Lik: -9432.134, Max-Change: 0.00686
Iteration: 13, Log-Lik: -9432.146, Max-Change: 0.00631
Iteration: 14, Log-Lik: -9432.174, Max-Change: 0.00277
Iteration: 15, Log-Lik: -9432.189, Max-Change: 0.00211
Iteration: 16, Log-Lik: -9432.197, Max-Change: 0.00185
Iteration: 17, Log-Lik: -9432.208, Max-Change: 0.00085
Iteration: 18, Log-Lik: -9432.213, Max-Change: 0.00067
Iteration: 19, Log-Lik: -9432.216, Max-Change: 0.00065
Iteration: 20, Log-Lik: -9432.219, Max-Change: 0.00016
Iteration: 21, Log-Lik: -9432.220, Max-Change: 0.00015
Iteration: 22, Log-Lik: -9432.221, Max-Change: 0.00013
Iteration: 23, Log-Lik: -9432.221, Max-Change: 0.00008
plot(dav, type = 'Davidian') # shape of latent trait distribution

coef(dav, simplify=TRUE)
#> $items
#>            a1      d g u
#> Item.1  0.774 -1.048 0 1
#> Item.2  1.684  0.495 0 1
#> Item.3  1.051 -1.114 0 1
#> Item.4  0.582 -0.531 0 1
#> Item.5  1.043  0.613 0 1
#> Item.6  1.037 -2.030 0 1
#> Item.7  1.096  1.397 0 1
#> Item.8  0.639 -1.513 0 1
#> Item.9  0.543  2.128 0 1
#> Item.10 0.993 -0.352 0 1
#> Item.11 2.130  5.453 0 1
#> Item.12 0.163 -0.338 0 1
#> Item.13 1.204  0.867 0 1
#> Item.14 1.171  1.211 0 1
#> Item.15 1.387  1.925 0 1
#> Item.16 0.725 -0.389 0 1
#> Item.17 1.860  4.273 0 1
#> Item.18 1.763 -0.788 0 1
#> Item.19 0.880  0.236 0 1
#> Item.20 1.866  2.743 0 1
#> Item.21 0.695  2.552 0 1
#> Item.22 1.863  3.592 0 1
#> Item.23 0.590 -0.851 0 1
#> Item.24 1.335  1.296 0 1
#> Item.25 0.733 -0.558 0 1
#> Item.26 1.649 -0.125 0 1
#> Item.27 2.356  2.968 0 1
#> Item.28 1.060  0.184 0 1
#> Item.29 0.803 -0.742 0 1
#> Item.30 0.352 -0.241 0 1
#> Item.31 2.944  3.061 0 1
#> Item.32 0.169 -1.651 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
#> $Davidian_phis
#> [1]  1.289  0.086 -0.443  1.245
#> 

fs <- fscores(dav) # assume normal prior
fs2 <- fscores(dav, use_dentype_estimate=TRUE) # use Davidian estimated prior shape
head(cbind(fs, fs2))
#>               F1           F1
#> [1,]  2.66818540  3.599616034
#> [2,]  0.14648879  0.070501775
#> [3,]  0.06802365  0.004037417
#> [4,] -0.41577386 -0.426755059
#> [5,]  0.67027700  0.559830142
#> [6,]  0.45477422  0.353831282

itemfit(dav) # assume normal prior
#> Error: Only X2, G2, PV_Q1, PV_Q1*, infit, X2*, and X2*_df can be computed with missing data.
#>              Pass na.rm=TRUE to remove missing data row-wise
itemfit(dav, use_dentype_estimate=TRUE) # use Davidian estimated prior shape
#> Error: Only X2, G2, PV_Q1, PV_Q1*, infit, X2*, and X2*_df can be computed with missing data.
#>              Pass na.rm=TRUE to remove missing data row-wise

############
# Unfolding models

# polytomous hyperbolic cosine model with
#  estimated latitude of acceptance (rho parameters)
mod <- mirt(Science, model=1, itemtype = 'hcm')
#> 
Iteration: 1, Log-Lik: -2887.501, Max-Change: 21.39161
Iteration: 2, Log-Lik: -1855.751, Max-Change: 0.74095
Iteration: 3, Log-Lik: -1832.969, Max-Change: 3.92655
Iteration: 4, Log-Lik: -1676.140, Max-Change: 0.87523
Iteration: 5, Log-Lik: -1641.089, Max-Change: 0.52198
Iteration: 6, Log-Lik: -1630.678, Max-Change: 0.15458
Iteration: 7, Log-Lik: -1628.627, Max-Change: 0.06557
Iteration: 8, Log-Lik: -1628.001, Max-Change: 0.03298
Iteration: 9, Log-Lik: -1627.778, Max-Change: 0.01721
Iteration: 10, Log-Lik: -1627.682, Max-Change: 0.00828
Iteration: 11, Log-Lik: -1627.662, Max-Change: 0.00447
Iteration: 12, Log-Lik: -1627.655, Max-Change: 0.00257
Iteration: 13, Log-Lik: -1627.652, Max-Change: 0.00146
Iteration: 14, Log-Lik: -1627.651, Max-Change: 0.00037
Iteration: 15, Log-Lik: -1627.651, Max-Change: 0.00048
Iteration: 16, Log-Lik: -1627.651, Max-Change: 0.00032
Iteration: 17, Log-Lik: -1627.651, Max-Change: 0.00015
Iteration: 18, Log-Lik: -1627.651, Max-Change: 0.00006
coef(mod, simplify=TRUE)$items
#>         a1        d log_rho1 log_rho2  log_rho3
#> Comfort  1 1.989655 1.625421 1.526102 -16.17198
#> Work     1 2.485054 1.478531 1.224489 -20.39161
#> Future   1 1.801569 1.491945 1.185052 -15.30324
#> Benefit  1 1.903674 1.471276 1.039336 -17.53716
coef(mod, simplify=TRUE, IRTpars=TRUE)$items
#>         a         b     rho1     rho2         rho3
#> Comfort 1 -1.989655 5.080557 4.600212 9.475379e-08
#> Work    1 -2.485054 4.386497 3.402426 1.393270e-09
#> Future  1 -1.801569 4.445732 3.270856 2.258856e-07
#> Benefit 1 -1.903674 4.354789 2.827339 2.419408e-08

plot(mod)

plot(mod, type = 'trace')

plot(mod, type = 'itemscore')


# EAP estimates
fs <- fscores(mod)
head(fs)
#>              F1
#> [1,] -0.6675994
#> [2,] -0.1584800
#> [3,]  0.6560938
#> [4,]  0.6560938
#> [5,] -0.2058628
#> [6,] -1.0079077

itemfit(mod)
#>      item   S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
#> 1 Comfort  4.437       6      0.000  0.618
#> 2    Work 10.778       8      0.030  0.215
#> 3  Future 18.242      10      0.046  0.051
#> 4 Benefit 12.020      11      0.015  0.362
M2(mod, type = 'C2')
#>             M2 df            p     RMSEA    RMSEA_5 RMSEA_95      TLI       CFI
#> stats 18.78276  2 8.344004e-05 0.1464969 0.09063855 0.209979 0.736325 0.9121083

###########
# 5PL and restricted 5PL example
dat <- expand.table(LSAT7)

mod2PL <- mirt(dat)
#> 
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2659.430, Max-Change: 0.04181
Iteration: 5, Log-Lik: -2659.241, Max-Change: 0.03417
Iteration: 6, Log-Lik: -2659.113, Max-Change: 0.02911
Iteration: 7, Log-Lik: -2658.812, Max-Change: 0.00456
Iteration: 8, Log-Lik: -2658.809, Max-Change: 0.00363
Iteration: 9, Log-Lik: -2658.808, Max-Change: 0.00273
Iteration: 10, Log-Lik: -2658.806, Max-Change: 0.00144
Iteration: 11, Log-Lik: -2658.806, Max-Change: 0.00118
Iteration: 12, Log-Lik: -2658.806, Max-Change: 0.00101
Iteration: 13, Log-Lik: -2658.805, Max-Change: 0.00042
Iteration: 14, Log-Lik: -2658.805, Max-Change: 0.00025
Iteration: 15, Log-Lik: -2658.805, Max-Change: 0.00026
Iteration: 16, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 17, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 18, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 19, Log-Lik: -2658.805, Max-Change: 0.00019
Iteration: 20, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 21, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 22, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 23, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 24, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 25, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 26, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 27, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 28, Log-Lik: -2658.805, Max-Change: 0.00010
mod2PL
#> 
#> Call:
#> mirt(data = dat)
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 28 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -2658.805
#> Estimated parameters: 10 
#> AIC = 5337.61
#> BIC = 5386.688; SABIC = 5354.927
#> G2 (21) = 31.7, p = 0.0628
#> RMSEA = 0.023, CFI = NaN, TLI = NaN

# Following does not converge without including strong priors
# mod5PL <- mirt(dat, itemtype = '5PL')
# mod5PL

# restricted version of 5PL (asymmetric 2PL)
model <- 'Theta = 1-5
          FIXED = (1-5, g), (1-5, u)'

mod2PL_asym <- mirt(dat, model=model, itemtype = '5PL')
#> 
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.51326
Iteration: 2, Log-Lik: -2663.511, Max-Change: 0.13862
Iteration: 3, Log-Lik: -2661.246, Max-Change: 0.08865
Iteration: 4, Log-Lik: -2659.548, Max-Change: 0.04834
Iteration: 5, Log-Lik: -2659.140, Max-Change: 0.08108
Iteration: 6, Log-Lik: -2658.895, Max-Change: 0.08582
Iteration: 7, Log-Lik: -2658.338, Max-Change: 0.03611
Iteration: 8, Log-Lik: -2658.230, Max-Change: 0.01253
Iteration: 9, Log-Lik: -2658.215, Max-Change: 0.04774
Iteration: 10, Log-Lik: -2658.190, Max-Change: 0.02676
Iteration: 11, Log-Lik: -2658.180, Max-Change: 0.05271
Iteration: 12, Log-Lik: -2658.164, Max-Change: 0.04346
Iteration: 13, Log-Lik: -2658.107, Max-Change: 0.05266
Iteration: 14, Log-Lik: -2658.089, Max-Change: 0.02992
Iteration: 15, Log-Lik: -2658.081, Max-Change: 0.04333
Iteration: 16, Log-Lik: -2658.049, Max-Change: 0.03464
Iteration: 17, Log-Lik: -2658.039, Max-Change: 0.02877
Iteration: 18, Log-Lik: -2658.032, Max-Change: 0.02668
Iteration: 19, Log-Lik: -2657.997, Max-Change: 0.02270
Iteration: 20, Log-Lik: -2657.991, Max-Change: 0.02926
Iteration: 21, Log-Lik: -2657.984, Max-Change: 0.01128
Iteration: 22, Log-Lik: -2657.983, Max-Change: 0.02726
Iteration: 23, Log-Lik: -2657.978, Max-Change: 0.02501
Iteration: 24, Log-Lik: -2657.972, Max-Change: 0.01968
Iteration: 25, Log-Lik: -2657.958, Max-Change: 0.00845
Iteration: 26, Log-Lik: -2657.956, Max-Change: 0.02232
Iteration: 27, Log-Lik: -2657.952, Max-Change: 0.01347
Iteration: 28, Log-Lik: -2657.947, Max-Change: 0.02663
Iteration: 29, Log-Lik: -2657.942, Max-Change: 0.01681
Iteration: 30, Log-Lik: -2657.938, Max-Change: 0.00044
Iteration: 31, Log-Lik: -2657.938, Max-Change: 0.00042
Iteration: 32, Log-Lik: -2657.938, Max-Change: 0.00040
Iteration: 33, Log-Lik: -2657.938, Max-Change: 0.00033
Iteration: 34, Log-Lik: -2657.938, Max-Change: 0.01223
Iteration: 35, Log-Lik: -2657.936, Max-Change: 0.01815
Iteration: 36, Log-Lik: -2657.933, Max-Change: 0.00064
Iteration: 37, Log-Lik: -2657.933, Max-Change: 0.01543
Iteration: 38, Log-Lik: -2657.930, Max-Change: 0.00043
Iteration: 39, Log-Lik: -2657.930, Max-Change: 0.00046
Iteration: 40, Log-Lik: -2657.930, Max-Change: 0.02086
Iteration: 41, Log-Lik: -2657.926, Max-Change: 0.00069
Iteration: 42, Log-Lik: -2657.926, Max-Change: 0.00062
Iteration: 43, Log-Lik: -2657.926, Max-Change: 0.00022
Iteration: 44, Log-Lik: -2657.926, Max-Change: 0.00014
Iteration: 45, Log-Lik: -2657.926, Max-Change: 0.00020
Iteration: 46, Log-Lik: -2657.926, Max-Change: 0.00623
Iteration: 47, Log-Lik: -2657.924, Max-Change: 0.00034
Iteration: 48, Log-Lik: -2657.924, Max-Change: 0.00025
Iteration: 49, Log-Lik: -2657.924, Max-Change: 0.00437
Iteration: 50, Log-Lik: -2657.923, Max-Change: 0.00087
Iteration: 51, Log-Lik: -2657.923, Max-Change: 0.00092
Iteration: 52, Log-Lik: -2657.923, Max-Change: 0.02021
Iteration: 53, Log-Lik: -2657.920, Max-Change: 0.00026
Iteration: 54, Log-Lik: -2657.920, Max-Change: 0.00015
Iteration: 55, Log-Lik: -2657.920, Max-Change: 0.00770
Iteration: 56, Log-Lik: -2657.919, Max-Change: 0.00205
Iteration: 57, Log-Lik: -2657.918, Max-Change: 0.00056
Iteration: 58, Log-Lik: -2657.918, Max-Change: 0.00031
Iteration: 59, Log-Lik: -2657.918, Max-Change: 0.00033
Iteration: 60, Log-Lik: -2657.918, Max-Change: 0.00029
Iteration: 61, Log-Lik: -2657.918, Max-Change: 0.01647
Iteration: 62, Log-Lik: -2657.915, Max-Change: 0.01720
Iteration: 63, Log-Lik: -2657.912, Max-Change: 0.00028
Iteration: 64, Log-Lik: -2657.912, Max-Change: 0.00027
Iteration: 65, Log-Lik: -2657.912, Max-Change: 0.00020
Iteration: 66, Log-Lik: -2657.912, Max-Change: 0.00028
Iteration: 67, Log-Lik: -2657.912, Max-Change: 0.00640
Iteration: 68, Log-Lik: -2657.911, Max-Change: 0.00070
Iteration: 69, Log-Lik: -2657.911, Max-Change: 0.00010
mod2PL_asym
#> 
#> Call:
#> mirt(data = dat, model = model, itemtype = "5PL")
#> 
#> Full-information item factor analysis with 1 factor(s).
#> Converged within 1e-04 tolerance after 69 EM iterations.
#> mirt version: 1.44.3 
#> M-step optimizer: BFGS 
#> EM acceleration: Ramsay 
#> Number of rectangular quadrature: 61
#> Latent density type: Gaussian 
#> 
#> Log-likelihood = -2657.911
#> Estimated parameters: 15 
#> AIC = 5345.822
#> BIC = 5419.438; SABIC = 5371.797
#> G2 (16) = 29.91, p = 0.0185
#> RMSEA = 0.03, CFI = NaN, TLI = NaN
coef(mod2PL_asym, simplify=TRUE)
#> $items
#>           a1      d g u   logS
#> Item.1 0.926  2.882 0 1  0.962
#> Item.2 2.141 -1.547 0 1 -1.454
#> Item.3 1.589  2.066 0 1  0.264
#> Item.4 0.613  2.223 0 1  1.518
#> Item.5 0.748  1.948 0 1  0.079
#> 
#> $means
#> Theta 
#>     0 
#> 
#> $cov
#>       Theta
#> Theta     1
#> 
coef(mod2PL_asym, simplify=TRUE, IRTpars=TRUE)
#> $items
#>            a      b g u     S
#> Item.1 0.926 -3.113 0 1 2.618
#> Item.2 2.141  0.722 0 1 0.234
#> Item.3 1.589 -1.301 0 1 1.301
#> Item.4 0.613 -3.627 0 1 4.563
#> Item.5 0.748 -2.605 0 1 1.082
#> 
#> $means
#> Theta 
#>     0 
#> 
#> $cov
#>       Theta
#> Theta     1
#> 

# no big difference statistically or visually
anova(mod2PL, mod2PL_asym)
#>                  AIC    SABIC       HQ      BIC    logLik    X2 df     p
#> mod2PL      5337.610 5354.927 5356.263 5386.688 -2658.805               
#> mod2PL_asym 5345.822 5371.797 5373.801 5419.438 -2657.911 1.788  5 0.878
plot(mod2PL, type = 'trace')

plot(mod2PL_asym, type = 'trace')



###################
# LLTM example

a <- matrix(rep(1,30))
d <- rep(c(1,0, -1),each = 10)  # first easy, then medium, last difficult
dat <- simdata(a, d, 1000, itemtype = '2PL')

# unconditional model for intercept comparisons
mod <- mirt(dat, itemtype = 'Rasch')
#> 
Iteration: 1, Log-Lik: -17542.943, Max-Change: 0.01620
Iteration: 2, Log-Lik: -17542.403, Max-Change: 0.01087
Iteration: 3, Log-Lik: -17542.353, Max-Change: 0.00443
Iteration: 4, Log-Lik: -17542.345, Max-Change: 0.00175
Iteration: 5, Log-Lik: -17542.344, Max-Change: 0.00080
Iteration: 6, Log-Lik: -17542.343, Max-Change: 0.00030
Iteration: 7, Log-Lik: -17542.343, Max-Change: 0.00015
Iteration: 8, Log-Lik: -17542.343, Max-Change: 0.00007
coef(mod, simplify=TRUE)
#> $items
#>         a1      d g u
#> Item_1   1  1.013 0 1
#> Item_2   1  1.181 0 1
#> Item_3   1  1.181 0 1
#> Item_4   1  1.087 0 1
#> Item_5   1  1.265 0 1
#> Item_6   1  0.936 0 1
#> Item_7   1  1.128 0 1
#> Item_8   1  1.076 0 1
#> Item_9   1  1.076 0 1
#> Item_10  1  1.187 0 1
#> Item_11  1  0.179 0 1
#> Item_12  1  0.106 0 1
#> Item_13  1 -0.014 0 1
#> Item_14  1  0.116 0 1
#> Item_15  1 -0.004 0 1
#> Item_16  1 -0.004 0 1
#> Item_17  1 -0.014 0 1
#> Item_18  1  0.097 0 1
#> Item_19  1  0.053 0 1
#> Item_20  1  0.005 0 1
#> Item_21  1 -0.968 0 1
#> Item_22  1 -0.837 0 1
#> Item_23  1 -0.880 0 1
#> Item_24  1 -0.929 0 1
#> Item_25  1 -0.913 0 1
#> Item_26  1 -0.924 0 1
#> Item_27  1 -0.968 0 1
#> Item_28  1 -0.951 0 1
#> Item_29  1 -1.035 0 1
#> Item_30  1 -0.951 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>       F1
#> F1 0.965
#> 

# Suppose that the first 10 items were suspected to be easy, followed by 10 medium difficulty items,
# then finally the last 10 items are difficult,
# and we wish to test this item structure hypothesis (more intercept designs are possible
# by including more columns).
itemdesign <- data.frame(difficulty =
   factor(c(rep('easy', 10), rep('medium', 10), rep('hard', 10))))
rownames(itemdesign) <- colnames(dat)
itemdesign
#>         difficulty
#> Item_1        easy
#> Item_2        easy
#> Item_3        easy
#> Item_4        easy
#> Item_5        easy
#> Item_6        easy
#> Item_7        easy
#> Item_8        easy
#> Item_9        easy
#> Item_10       easy
#> Item_11     medium
#> Item_12     medium
#> Item_13     medium
#> Item_14     medium
#> Item_15     medium
#> Item_16     medium
#> Item_17     medium
#> Item_18     medium
#> Item_19     medium
#> Item_20     medium
#> Item_21       hard
#> Item_22       hard
#> Item_23       hard
#> Item_24       hard
#> Item_25       hard
#> Item_26       hard
#> Item_27       hard
#> Item_28       hard
#> Item_29       hard
#> Item_30       hard

# LLTM with mirt()
lltm <- mirt(dat, itemtype = 'Rasch', SE=TRUE,
   item.formula = ~ 0 + difficulty, itemdesign=itemdesign)
#> 
Iteration: 1, Log-Lik: -19541.318, Max-Change: 1.02541
Iteration: 2, Log-Lik: -17563.535, Max-Change: 0.08816
Iteration: 3, Log-Lik: -17558.308, Max-Change: 0.04162
Iteration: 4, Log-Lik: -17557.147, Max-Change: 0.01778
Iteration: 5, Log-Lik: -17556.796, Max-Change: 0.00723
Iteration: 6, Log-Lik: -17556.629, Max-Change: 0.00489
Iteration: 7, Log-Lik: -17556.322, Max-Change: 0.00270
Iteration: 8, Log-Lik: -17556.306, Max-Change: 0.00140
Iteration: 9, Log-Lik: -17556.296, Max-Change: 0.00118
Iteration: 10, Log-Lik: -17556.271, Max-Change: 0.00020
Iteration: 11, Log-Lik: -17556.271, Max-Change: 0.00016
Iteration: 12, Log-Lik: -17556.271, Max-Change: 0.00013
Iteration: 13, Log-Lik: -17556.271, Max-Change: 0.00002
#> 
#> Calculating information matrix...
coef(lltm, simplify=TRUE)
#> $items
#>         difficultyeasy difficultyhard difficultymedium a1 d g u
#> Item_1           1.111          0.000            0.000  1 0 0 1
#> Item_2           1.111          0.000            0.000  1 0 0 1
#> Item_3           1.111          0.000            0.000  1 0 0 1
#> Item_4           1.111          0.000            0.000  1 0 0 1
#> Item_5           1.111          0.000            0.000  1 0 0 1
#> Item_6           1.111          0.000            0.000  1 0 0 1
#> Item_7           1.111          0.000            0.000  1 0 0 1
#> Item_8           1.111          0.000            0.000  1 0 0 1
#> Item_9           1.111          0.000            0.000  1 0 0 1
#> Item_10          1.111          0.000            0.000  1 0 0 1
#> Item_11          0.000          0.000            0.052  1 0 0 1
#> Item_12          0.000          0.000            0.052  1 0 0 1
#> Item_13          0.000          0.000            0.052  1 0 0 1
#> Item_14          0.000          0.000            0.052  1 0 0 1
#> Item_15          0.000          0.000            0.052  1 0 0 1
#> Item_16          0.000          0.000            0.052  1 0 0 1
#> Item_17          0.000          0.000            0.052  1 0 0 1
#> Item_18          0.000          0.000            0.052  1 0 0 1
#> Item_19          0.000          0.000            0.052  1 0 0 1
#> Item_20          0.000          0.000            0.052  1 0 0 1
#> Item_21          0.000         -0.935            0.000  1 0 0 1
#> Item_22          0.000         -0.935            0.000  1 0 0 1
#> Item_23          0.000         -0.935            0.000  1 0 0 1
#> Item_24          0.000         -0.935            0.000  1 0 0 1
#> Item_25          0.000         -0.935            0.000  1 0 0 1
#> Item_26          0.000         -0.935            0.000  1 0 0 1
#> Item_27          0.000         -0.935            0.000  1 0 0 1
#> Item_28          0.000         -0.935            0.000  1 0 0 1
#> Item_29          0.000         -0.935            0.000  1 0 0 1
#> Item_30          0.000         -0.935            0.000  1 0 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>       F1
#> F1 0.963
#> 
coef(lltm, printSE=TRUE)
#> $Item_1
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_2
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_3
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_4
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_5
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_6
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_7
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_8
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_9
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_10
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_11
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_12
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_13
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_14
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_15
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_16
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_17
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_18
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_19
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_20
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_21
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_22
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_23
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_24
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_25
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_26
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_27
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_28
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_29
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $Item_30
#>     difficultyeasy difficultyhard difficultymedium a1  d logit(g) logit(u)
#> par          1.111         -0.935            0.052  1  0     -999      999
#> SE           0.040          0.039            0.038 NA NA       NA       NA
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0  0.963
#> SE      NA  0.055
#> 
anova(lltm, mod)  # models fit effectively the same; hence, intercept variability well captured
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> lltm 35120.54 35127.47 35128.00 35140.17 -17556.27                
#> mod  35146.69 35200.37 35204.51 35298.83 -17542.34 27.855 27 0.418

# additional information for LLTM
plot(lltm)

plot(lltm, type = 'trace')

itemplot(lltm, item=1)

itemfit(lltm)
#>       item   S_X2 df.S_X2 RMSEA.S_X2 p.S_X2
#> 1   Item_1 23.752      20      0.014  0.253
#> 2   Item_2 23.130      20      0.013  0.282
#> 3   Item_3 27.993      20      0.020  0.110
#> 4   Item_4 40.861      20      0.032  0.004
#> 5   Item_5 31.981      20      0.024  0.044
#> 6   Item_6 23.488      20      0.013  0.265
#> 7   Item_7 43.655      20      0.034  0.002
#> 8   Item_8 21.861      20      0.010  0.348
#> 9   Item_9 21.998      20      0.010  0.341
#> 10 Item_10 37.390      20      0.030  0.011
#> 11 Item_11 28.113      21      0.018  0.137
#> 12 Item_12 28.969      21      0.019  0.115
#> 13 Item_13 20.205      21      0.000  0.508
#> 14 Item_14 43.886      21      0.033  0.002
#> 15 Item_15 14.443      21      0.000  0.850
#> 16 Item_16 28.846      21      0.019  0.118
#> 17 Item_17 21.570      21      0.005  0.425
#> 18 Item_18 20.712      21      0.000  0.477
#> 19 Item_19 34.553      21      0.025  0.032
#> 20 Item_20 29.787      21      0.020  0.096
#> 21 Item_21 30.672      20      0.023  0.060
#> 22 Item_22 25.632      20      0.017  0.178
#> 23 Item_23 21.496      20      0.009  0.368
#> 24 Item_24 25.820      20      0.017  0.172
#> 25 Item_25 15.185      20      0.000  0.766
#> 26 Item_26 20.141      20      0.003  0.449
#> 27 Item_27 34.447      20      0.027  0.023
#> 28 Item_28 35.993      20      0.028  0.015
#> 29 Item_29 26.583      20      0.018  0.147
#> 30 Item_30 17.789      20      0.000  0.601
head(fscores(lltm))  #EAP estimates
#>              F1
#> [1,] -1.0635660
#> [2,]  0.2114673
#> [3,]  1.0975084
#> [4,] -0.6186294
#> [5,]  0.2114673
#> [6,]  0.4919959
fscores(lltm, method='EAPsum', full.scores=FALSE)
#>       df     X2  p.X2 SEM.alpha rxx.alpha rxx_F1
#> stats 29 24.235 0.717     2.379     0.844  0.841
#> 
#>    Sum.Scores     F1 SE_F1 observed expected std.res
#> 0           0 -2.718 0.552        1    0.770   0.262
#> 1           1 -2.433 0.515        2    2.637   0.393
#> 2           2 -2.183 0.486        5    5.585   0.247
#> 3           3 -1.959 0.462       11    9.450   0.504
#> 4           4 -1.755 0.443       14   14.026   0.007
#> 5           5 -1.566 0.427       15   19.100   0.938
#> 6           6 -1.389 0.414       24   24.468   0.095
#> 7           7 -1.223 0.403       31   29.937   0.194
#> 8           8 -1.064 0.395       47   35.328   1.964
#> 9           9 -0.911 0.388       41   40.472   0.083
#> 10         10 -0.763 0.382       35   45.218   1.519
#> 11         11 -0.619 0.378       53   49.426   0.508
#> 12         12 -0.477 0.374       50   52.977   0.409
#> 13         13 -0.338 0.372       55   55.769   0.103
#> 14         14 -0.201 0.371       57   57.721   0.095
#> 15         15 -0.063 0.370       45   58.777   1.797
#> 16         16  0.074 0.371       74   58.900   1.967
#> 17         17  0.211 0.372       61   58.081   0.383
#> 18         18  0.351 0.374       59   56.332   0.355
#> 19         19  0.492 0.378       48   53.691   0.777
#> 20         20  0.636 0.382       47   50.218   0.454
#> 21         21  0.785 0.388       55   45.998   1.327
#> 22         22  0.938 0.395       41   41.137   0.021
#> 23         23  1.098 0.404       32   35.769   0.630
#> 24         24  1.265 0.415       29   30.047   0.191
#> 25         25  1.443 0.428       26   24.157   0.375
#> 26         26  1.633 0.445       17   18.316   0.308
#> 27         27  1.840 0.465       11   12.780   0.498
#> 28         28  2.067 0.490        5    7.851   1.018
#> 29         29  2.322 0.520        8    3.872   2.097
#> 30         30  2.612 0.558        1    1.189   0.173
M2(lltm) # goodness of fit
#>             M2  df         p       RMSEA RMSEA_5   RMSEA_95      SRMSR
#> stats 487.6935 461 0.1882019 0.007613246       0 0.01347354 0.03179874
#>             TLI       CFI
#> stats 0.9975308 0.9973832
head(personfit(lltm))
#>      outfit   z.outfit     infit    z.infit         Zh
#> 1 1.1453002  0.6017489 1.0598868  0.3791985 -0.3978746
#> 2 0.9677996 -0.1331664 0.9687086 -0.1659070  0.2047321
#> 3 0.7530052 -0.7663314 0.8973446 -0.4655325  0.6124102
#> 4 0.8359489 -0.8138577 0.8611352 -0.8738968  0.8680327
#> 5 1.1084361  0.6690511 1.1199264  0.8336140 -0.7552087
#> 6 0.9078434 -0.4036949 0.9437998 -0.3180501  0.3975139
residuals(lltm)
#> LD matrix (lower triangle) and standardized residual correlations (upper triangle)
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.122  -0.044  -0.011   0.000   0.044   0.104 
#> 
#>         Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
#> Item_1          0.051  0.068  0.055  0.101  0.089 -0.063  0.039  0.057  -0.055
#> Item_2   2.575        -0.049  0.057  0.060 -0.081  0.037 -0.055  0.059   0.067
#> Item_3   4.666  2.408        -0.076 -0.060 -0.089  0.026  0.034 -0.036   0.036
#> Item_4   3.052  3.217  5.778         0.065  0.095 -0.015 -0.041  0.021   0.041
#> Item_5  10.290  3.643  3.606  4.166         0.098  0.058  0.062 -0.063   0.104
#> Item_6   7.940  6.518  7.967  8.980  9.557         0.071 -0.122  0.077   0.102
#> Item_7   3.965  1.397  0.700  0.222  3.411  5.082        -0.027  0.061   0.040
#> Item_8   1.537  3.018  1.152  1.643  3.794 14.780  0.717        -0.040  -0.045
#> Item_9   3.229  3.507  1.275  0.462  3.957  5.899  3.769  1.617          0.034
#> Item_10  2.985  4.535  1.322  1.692 10.766 10.378  1.604  1.982  1.140        
#> Item_11  7.124  4.522  3.204  3.466  5.495  9.288  3.165  3.504  3.380   3.472
#> Item_12  4.837  1.445  1.445  2.165  3.521  5.963  0.520  0.850  1.202   2.428
#> Item_13  6.404  3.245  1.716  3.940  8.348  5.097  1.721  1.105  3.105   1.916
#> Item_14  5.652  4.123  1.923  2.928  3.785  7.612  0.707  3.560  2.011   6.154
#> Item_15  1.777  1.465  1.478  2.193  4.356  5.945  1.692  0.742  0.742   2.404
#> Item_16  8.296  2.021  7.266  3.469  4.555  4.993  2.661  1.166  0.839   1.617
#> Item_17  2.042  1.709  2.910  1.050  5.042  5.676  3.163  3.579  1.998   2.216
#> Item_18  2.503  1.913  1.083  0.672  3.467  6.592  1.121  1.101  2.603   1.144
#> Item_19  2.038  0.950  2.556  1.930  3.394  4.959  0.080  5.245  3.321   1.580
#> Item_20  1.921  1.359  1.318  1.835  4.163  6.019  0.504  1.957  0.506   1.619
#> Item_21  1.521  0.968  1.015  0.288  3.809  4.845  0.250  0.556  0.313   1.889
#> Item_22  3.680  2.387  2.193  1.761  4.572 13.780  4.138  2.465  1.932   2.254
#> Item_23  3.828  5.080  1.497  0.621  3.864  5.939  0.661  3.812  1.104   1.125
#> Item_24  4.387  2.996  1.952  0.219  3.448  5.818  3.721  1.710  0.367   1.064
#> Item_25  2.616  1.358  0.936  1.435  4.450  6.235  0.767  2.160  0.796   0.841
#> Item_26  1.722  1.488  0.691  0.490  3.541  5.032  1.972  0.247  0.269   1.443
#> Item_27  1.521  2.554  1.274  0.391  4.977  8.675  0.269  0.420  1.656   1.167
#> Item_28  2.047  0.836  1.392  0.153  5.285  4.852  0.920  4.350  0.288   1.153
#> Item_29  5.337  2.621  7.830  2.797  5.679  7.602  2.107  1.722  5.080   2.667
#> Item_30  1.534  1.873  2.670  1.162  3.755  5.095  0.647  0.357  0.581   3.156
#>         Item_11 Item_12 Item_13 Item_14 Item_15 Item_16 Item_17 Item_18 Item_19
#> Item_1    0.084   0.070  -0.080  -0.075  -0.042   0.091   0.045   0.050   0.045
#> Item_2   -0.067  -0.038   0.057  -0.064   0.038   0.045  -0.041  -0.044  -0.031
#> Item_3   -0.057  -0.038   0.041  -0.044  -0.038  -0.085   0.054   0.033   0.051
#> Item_4   -0.059  -0.047  -0.063  -0.054  -0.047  -0.059   0.032   0.026   0.044
#> Item_5    0.074  -0.059  -0.091  -0.062   0.066   0.067  -0.071  -0.059   0.058
#> Item_6    0.096   0.077  -0.071  -0.087   0.077  -0.071   0.075   0.081   0.070
#> Item_7   -0.056  -0.023  -0.041  -0.027   0.041  -0.052   0.056  -0.033   0.009
#> Item_8   -0.059   0.029  -0.033  -0.060   0.027  -0.034  -0.060   0.033  -0.072
#> Item_9   -0.058   0.035   0.056   0.045   0.027  -0.029   0.045   0.051   0.058
#> Item_10  -0.059  -0.049  -0.044  -0.078  -0.049   0.040   0.047   0.034   0.040
#> Item_11          -0.056  -0.064  -0.080  -0.091  -0.073   0.064   0.090   0.055
#> Item_12   3.107           0.039  -0.082  -0.048  -0.043  -0.041  -0.044  -0.050
#> Item_13   4.152   1.516          -0.064  -0.044  -0.039  -0.040   0.037   0.059
#> Item_14   6.358   6.673   4.051          -0.040  -0.068   0.059  -0.032  -0.074
#> Item_15   8.278   2.277   1.956   1.563           0.031   0.037   0.047  -0.027
#> Item_16   5.297   1.829   1.553   4.655   0.958          -0.035  -0.040  -0.024
#> Item_17   4.122   1.643   1.571   3.470   1.372   1.196           0.049   0.066
#> Item_18   8.148   1.909   1.333   1.006   2.184   1.624   2.424           0.019
#> Item_19   3.031   2.473   3.483   5.431   0.744   0.597   4.393   0.350        
#> Item_20   4.297   1.623   1.058   3.228   1.576   1.084   1.763   4.677   2.306
#> Item_21   3.616   3.592   0.956   3.194   1.909   0.711   2.994   0.637   0.206
#> Item_22   4.499   1.978   2.771   4.947   3.954   3.954   2.810   1.675   4.127
#> Item_23   3.293   0.936   1.996   2.894   6.652   1.219   2.488   0.922   6.172
#> Item_24   3.200   0.529   0.805   0.759   0.675   0.658   1.040   0.750   0.439
#> Item_25   2.759   1.181   1.373   2.433   0.736   1.711   1.134   0.833   0.479
#> Item_26   3.506   0.558   1.555   0.765   0.884   0.884   0.857   0.521   0.379
#> Item_27   4.321   1.740   0.836   2.377   0.711   0.723   1.937   0.637   0.228
#> Item_28   3.010   2.095   2.072   6.019   0.842   4.274   1.043   1.968   0.174
#> Item_29   8.636   5.111   2.780   3.224   2.099   3.701   6.623   2.244   4.005
#> Item_30   3.626   1.045   1.663   1.372   0.713   1.309   0.824   0.593   0.294
#>         Item_20 Item_21 Item_22 Item_23 Item_24 Item_25 Item_26 Item_27 Item_28
#> Item_1   -0.044  -0.039  -0.061  -0.062   0.066   0.051   0.041  -0.039  -0.045
#> Item_2   -0.037  -0.031  -0.049   0.071   0.055   0.037   0.039   0.051   0.029
#> Item_3    0.036  -0.032   0.047   0.039  -0.044   0.031   0.026  -0.036  -0.037
#> Item_4   -0.043   0.017  -0.042   0.025   0.015   0.038  -0.022  -0.020   0.012
#> Item_5    0.065  -0.062  -0.068   0.062  -0.059  -0.067   0.060  -0.071  -0.073
#> Item_6    0.078   0.070  -0.117  -0.077   0.076   0.079  -0.071  -0.093  -0.070
#> Item_7   -0.022   0.016  -0.064  -0.026  -0.061  -0.028  -0.044  -0.016  -0.030
#> Item_8   -0.044  -0.024  -0.050   0.062  -0.041   0.046   0.016  -0.021  -0.066
#> Item_9    0.022  -0.018   0.044   0.033  -0.019   0.028  -0.016   0.041   0.017
#> Item_10   0.040   0.043   0.047  -0.034  -0.033   0.029  -0.038  -0.034  -0.034
#> Item_11   0.066  -0.060  -0.067   0.057  -0.057  -0.053  -0.059  -0.066  -0.055
#> Item_12  -0.040  -0.060   0.044  -0.031  -0.023  -0.034  -0.024  -0.042  -0.046
#> Item_13  -0.033   0.031  -0.053   0.045  -0.028   0.037  -0.039  -0.029  -0.046
#> Item_14  -0.057  -0.057  -0.070  -0.054  -0.028  -0.049   0.028   0.049  -0.078
#> Item_15   0.040  -0.044  -0.063   0.082  -0.026   0.027  -0.030   0.027  -0.029
#> Item_16  -0.033   0.027  -0.063   0.035   0.026   0.041  -0.030  -0.027  -0.065
#> Item_17   0.042   0.055   0.053   0.050  -0.032  -0.034  -0.029   0.044   0.032
#> Item_18   0.068  -0.025   0.041   0.030   0.027   0.029  -0.023  -0.025   0.044
#> Item_19   0.048   0.014  -0.064   0.079  -0.021   0.022  -0.019  -0.015   0.013
#> Item_20          -0.023   0.051  -0.044  -0.022   0.044   0.022  -0.023   0.025
#> Item_21   0.534          -0.061   0.040   0.016   0.035  -0.016  -0.017  -0.055
#> Item_22   2.575   3.682          -0.053   0.039   0.053  -0.039  -0.055  -0.066
#> Item_23   1.932   1.612   2.822           0.023   0.024  -0.026  -0.067  -0.024
#> Item_24   0.466   0.251   1.522   0.548          -0.009   0.023  -0.023   0.027
#> Item_25   1.951   1.209   2.822   0.568   0.090           0.024  -0.022   0.014
#> Item_26   0.494   0.257   1.544   0.702   0.513   0.558           0.041  -0.011
#> Item_27   0.534   0.296   2.993   4.465   0.546   0.505   1.672          -0.025
#> Item_28   0.644   3.018   4.371   0.570   0.723   0.191   0.127   0.648        
#> Item_29   2.036   3.911   8.176   3.908   1.667   4.721   4.012   1.611   1.510
#> Item_30   5.136   0.992   2.814   1.449   0.978   0.786   0.073   0.198   0.100
#>         Item_29 Item_30
#> Item_1    0.073  -0.039
#> Item_2    0.051   0.043
#> Item_3    0.088   0.052
#> Item_4    0.053   0.034
#> Item_5    0.075  -0.061
#> Item_6    0.087  -0.071
#> Item_7    0.046  -0.025
#> Item_8    0.041  -0.019
#> Item_9    0.071   0.024
#> Item_10  -0.052   0.056
#> Item_11  -0.093   0.060
#> Item_12   0.071  -0.032
#> Item_13   0.053   0.041
#> Item_14  -0.057  -0.037
#> Item_15   0.046  -0.027
#> Item_16   0.061   0.036
#> Item_17   0.081   0.029
#> Item_18  -0.047   0.024
#> Item_19   0.063   0.017
#> Item_20   0.045   0.072
#> Item_21   0.063   0.031
#> Item_22   0.090  -0.053
#> Item_23   0.063  -0.038
#> Item_24   0.041   0.031
#> Item_25   0.069   0.028
#> Item_26   0.063   0.009
#> Item_27   0.040  -0.014
#> Item_28   0.039   0.010
#> Item_29           0.044
#> Item_30   1.942        

# intercept across items also possible by removing ~ 0 portion, just interpreted differently
lltm.int <- mirt(dat, itemtype = 'Rasch',
   item.formula = ~ difficulty, itemdesign=itemdesign)
#> 
Iteration: 1, Log-Lik: -19541.318, Max-Change: 1.99267
Iteration: 2, Log-Lik: -17563.535, Max-Change: 0.08820
Iteration: 3, Log-Lik: -17558.307, Max-Change: 0.04160
Iteration: 4, Log-Lik: -17557.147, Max-Change: 0.01777
Iteration: 5, Log-Lik: -17556.796, Max-Change: 0.00721
Iteration: 6, Log-Lik: -17556.629, Max-Change: 0.00489
Iteration: 7, Log-Lik: -17556.421, Max-Change: 0.00251
Iteration: 8, Log-Lik: -17556.377, Max-Change: 0.00247
Iteration: 9, Log-Lik: -17556.347, Max-Change: 0.00206
Iteration: 10, Log-Lik: -17556.274, Max-Change: 0.00036
Iteration: 11, Log-Lik: -17556.273, Max-Change: 0.00035
Iteration: 12, Log-Lik: -17556.272, Max-Change: 0.00030
Iteration: 13, Log-Lik: -17556.271, Max-Change: 0.00004
anova(lltm, lltm.int) # same
#>               AIC    SABIC    HQ      BIC    logLik X2 df   p
#> lltm     35120.54 35127.47 35128 35140.17 -17556.27          
#> lltm.int 35120.54 35127.47 35128 35140.17 -17556.27  0  0 NaN
coef(lltm.int, simplify=TRUE)
#> $items
#>         (Intercept) difficultyhard difficultymedium a1 d g u
#> Item_1        1.111          0.000            0.000  1 0 0 1
#> Item_2        1.111          0.000            0.000  1 0 0 1
#> Item_3        1.111          0.000            0.000  1 0 0 1
#> Item_4        1.111          0.000            0.000  1 0 0 1
#> Item_5        1.111          0.000            0.000  1 0 0 1
#> Item_6        1.111          0.000            0.000  1 0 0 1
#> Item_7        1.111          0.000            0.000  1 0 0 1
#> Item_8        1.111          0.000            0.000  1 0 0 1
#> Item_9        1.111          0.000            0.000  1 0 0 1
#> Item_10       1.111          0.000            0.000  1 0 0 1
#> Item_11       1.111          0.000           -1.059  1 0 0 1
#> Item_12       1.111          0.000           -1.059  1 0 0 1
#> Item_13       1.111          0.000           -1.059  1 0 0 1
#> Item_14       1.111          0.000           -1.059  1 0 0 1
#> Item_15       1.111          0.000           -1.059  1 0 0 1
#> Item_16       1.111          0.000           -1.059  1 0 0 1
#> Item_17       1.111          0.000           -1.059  1 0 0 1
#> Item_18       1.111          0.000           -1.059  1 0 0 1
#> Item_19       1.111          0.000           -1.059  1 0 0 1
#> Item_20       1.111          0.000           -1.059  1 0 0 1
#> Item_21       1.111         -2.046            0.000  1 0 0 1
#> Item_22       1.111         -2.046            0.000  1 0 0 1
#> Item_23       1.111         -2.046            0.000  1 0 0 1
#> Item_24       1.111         -2.046            0.000  1 0 0 1
#> Item_25       1.111         -2.046            0.000  1 0 0 1
#> Item_26       1.111         -2.046            0.000  1 0 0 1
#> Item_27       1.111         -2.046            0.000  1 0 0 1
#> Item_28       1.111         -2.046            0.000  1 0 0 1
#> Item_29       1.111         -2.046            0.000  1 0 0 1
#> Item_30       1.111         -2.046            0.000  1 0 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>       F1
#> F1 0.963
#> 

# using unconditional modeling for first four items
itemdesign.sub <- itemdesign[5:nrow(itemdesign), , drop=FALSE]
itemdesign.sub    # note that rownames are required in this case
#>         difficulty
#> Item_5        easy
#> Item_6        easy
#> Item_7        easy
#> Item_8        easy
#> Item_9        easy
#> Item_10       easy
#> Item_11     medium
#> Item_12     medium
#> Item_13     medium
#> Item_14     medium
#> Item_15     medium
#> Item_16     medium
#> Item_17     medium
#> Item_18     medium
#> Item_19     medium
#> Item_20     medium
#> Item_21       hard
#> Item_22       hard
#> Item_23       hard
#> Item_24       hard
#> Item_25       hard
#> Item_26       hard
#> Item_27       hard
#> Item_28       hard
#> Item_29       hard
#> Item_30       hard
lltm.4 <- mirt(dat, itemtype = 'Rasch',
   item.formula = ~ 0 + difficulty, itemdesign=itemdesign.sub)
#> 
Iteration: 1, Log-Lik: -19076.961, Max-Change: 1.15268
Iteration: 2, Log-Lik: -17559.895, Max-Change: 0.07099
Iteration: 3, Log-Lik: -17556.412, Max-Change: 0.03204
Iteration: 4, Log-Lik: -17555.460, Max-Change: 0.01334
Iteration: 5, Log-Lik: -17555.147, Max-Change: 0.00643
Iteration: 6, Log-Lik: -17554.970, Max-Change: 0.00506
Iteration: 7, Log-Lik: -17554.588, Max-Change: 0.00226
Iteration: 8, Log-Lik: -17554.578, Max-Change: 0.00114
Iteration: 9, Log-Lik: -17554.571, Max-Change: 0.00095
Iteration: 10, Log-Lik: -17554.556, Max-Change: 0.00020
Iteration: 11, Log-Lik: -17554.556, Max-Change: 0.00013
Iteration: 12, Log-Lik: -17554.556, Max-Change: 0.00011
Iteration: 13, Log-Lik: -17554.555, Max-Change: 0.00002
coef(lltm.4, simplify=TRUE) # first four items are the standard Rasch
#> $items
#>         difficultyeasy difficultyhard difficultymedium a1     d g u
#> Item_1           0.000          0.000            0.000  1 1.013 0 1
#> Item_2           0.000          0.000            0.000  1 1.180 0 1
#> Item_3           0.000          0.000            0.000  1 1.180 0 1
#> Item_4           0.000          0.000            0.000  1 1.087 0 1
#> Item_5           1.109          0.000            0.000  1 0.000 0 1
#> Item_6           1.109          0.000            0.000  1 0.000 0 1
#> Item_7           1.109          0.000            0.000  1 0.000 0 1
#> Item_8           1.109          0.000            0.000  1 0.000 0 1
#> Item_9           1.109          0.000            0.000  1 0.000 0 1
#> Item_10          1.109          0.000            0.000  1 0.000 0 1
#> Item_11          0.000          0.000            0.052  1 0.000 0 1
#> Item_12          0.000          0.000            0.052  1 0.000 0 1
#> Item_13          0.000          0.000            0.052  1 0.000 0 1
#> Item_14          0.000          0.000            0.052  1 0.000 0 1
#> Item_15          0.000          0.000            0.052  1 0.000 0 1
#> Item_16          0.000          0.000            0.052  1 0.000 0 1
#> Item_17          0.000          0.000            0.052  1 0.000 0 1
#> Item_18          0.000          0.000            0.052  1 0.000 0 1
#> Item_19          0.000          0.000            0.052  1 0.000 0 1
#> Item_20          0.000          0.000            0.052  1 0.000 0 1
#> Item_21          0.000         -0.935            0.000  1 0.000 0 1
#> Item_22          0.000         -0.935            0.000  1 0.000 0 1
#> Item_23          0.000         -0.935            0.000  1 0.000 0 1
#> Item_24          0.000         -0.935            0.000  1 0.000 0 1
#> Item_25          0.000         -0.935            0.000  1 0.000 0 1
#> Item_26          0.000         -0.935            0.000  1 0.000 0 1
#> Item_27          0.000         -0.935            0.000  1 0.000 0 1
#> Item_28          0.000         -0.935            0.000  1 0.000 0 1
#> Item_29          0.000         -0.935            0.000  1 0.000 0 1
#> Item_30          0.000         -0.935            0.000  1 0.000 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>       F1
#> F1 0.963
#> 
anova(lltm, lltm.4) # similar fit, hence more constrained model preferred
#>             AIC    SABIC       HQ      BIC    logLik    X2 df     p
#> lltm   35120.54 35127.47 35128.00 35140.17 -17556.27               
#> lltm.4 35125.11 35138.96 35140.03 35164.37 -17554.56 3.431  4 0.488

# LLTM with mixedmirt() (more flexible in general, but slower)
LLTM <- mixedmirt(dat, model=1, fixed = ~ 0 + difficulty,
                  itemdesign=itemdesign, SE=FALSE)
#> 
Stage 1 = 1, CDLL = -19926.5, AR(0.55) = [0.51], Max-Change = 0.1895
Stage 1 = 2, CDLL = -19273.2, AR(0.55) = [0.51], Max-Change = 0.1532
Stage 1 = 3, CDLL = -18887.8, AR(0.55) = [0.50], Max-Change = 0.1314
Stage 1 = 4, CDLL = -18569.4, AR(0.55) = [0.47], Max-Change = 0.1007
Stage 1 = 5, CDLL = -18373.7, AR(0.55) = [0.48], Max-Change = 0.0870
Stage 1 = 6, CDLL = -18243.0, AR(0.55) = [0.48], Max-Change = 0.0715
Stage 1 = 7, CDLL = -18109.2, AR(0.55) = [0.49], Max-Change = 0.0569
Stage 1 = 8, CDLL = -18135.3, AR(0.55) = [0.53], Max-Change = 0.0476
Stage 1 = 9, CDLL = -18055.8, AR(0.55) = [0.53], Max-Change = 0.0393
Stage 1 = 10, CDLL = -18031.8, AR(0.55) = [0.50], Max-Change = 0.0356
Stage 1 = 11, CDLL = -18010.3, AR(0.55) = [0.52], Max-Change = 0.0282
Stage 1 = 12, CDLL = -18023.4, AR(0.55) = [0.48], Max-Change = 0.0228
Stage 1 = 13, CDLL = -18031.6, AR(0.55) = [0.53], Max-Change = 0.0195
Stage 1 = 14, CDLL = -18037.1, AR(0.55) = [0.49], Max-Change = 0.0186
Stage 1 = 15, CDLL = -18042.3, AR(0.55) = [0.51], Max-Change = 0.0167
Stage 1 = 16, CDLL = -18021.7, AR(0.55) = [0.51], Max-Change = 0.0112
Stage 1 = 17, CDLL = -18015.7, AR(0.55) = [0.51], Max-Change = 0.0157
Stage 1 = 18, CDLL = -18040.2, AR(0.55) = [0.52], Max-Change = 0.0056
Stage 1 = 19, CDLL = -18055.3, AR(0.55) = [0.49], Max-Change = 0.0061
Stage 1 = 20, CDLL = -18031.5, AR(0.55) = [0.50], Max-Change = 0.0057
Stage 1 = 21, CDLL = -18005.0, AR(0.55) = [0.49], Max-Change = 0.0061
Stage 1 = 22, CDLL = -18014.5, AR(0.55) = [0.51], Max-Change = 0.0064
Stage 1 = 23, CDLL = -18035.0, AR(0.55) = [0.53], Max-Change = 0.0048
Stage 1 = 24, CDLL = -18050.3, AR(0.55) = [0.52], Max-Change = 0.0053
Stage 1 = 25, CDLL = -17996.4, AR(0.55) = [0.50], Max-Change = 0.0050
Stage 1 = 26, CDLL = -18021.3, AR(0.55) = [0.51], Max-Change = 0.0128
Stage 1 = 27, CDLL = -18074.3, AR(0.55) = [0.55], Max-Change = 0.0070
Stage 1 = 28, CDLL = -18016.5, AR(0.55) = [0.52], Max-Change = 0.0037
Stage 1 = 29, CDLL = -18022.5, AR(0.55) = [0.49], Max-Change = 0.0043
Stage 1 = 30, CDLL = -18031.9, AR(0.55) = [0.53], Max-Change = 0.0089
Stage 1 = 31, CDLL = -18024.6, AR(0.55) = [0.51], Max-Change = 0.0025
Stage 1 = 32, CDLL = -18024.2, AR(0.55) = [0.51], Max-Change = 0.0037
Stage 1 = 33, CDLL = -17984.2, AR(0.55) = [0.52], Max-Change = 0.0030
Stage 1 = 34, CDLL = -18003.4, AR(0.55) = [0.50], Max-Change = 0.0023
Stage 1 = 35, CDLL = -18014.3, AR(0.55) = [0.51], Max-Change = 0.0061
Stage 1 = 36, CDLL = -18046.5, AR(0.55) = [0.53], Max-Change = 0.0037
Stage 1 = 37, CDLL = -18019.2, AR(0.55) = [0.54], Max-Change = 0.0037
Stage 1 = 38, CDLL = -18020.1, AR(0.55) = [0.53], Max-Change = 0.0008
Stage 1 = 39, CDLL = -18072.9, AR(0.55) = [0.52], Max-Change = 0.0058
Stage 1 = 40, CDLL = -18016.6, AR(0.55) = [0.51], Max-Change = 0.0078
Stage 1 = 41, CDLL = -18050.3, AR(0.55) = [0.52], Max-Change = 0.0026
Stage 1 = 42, CDLL = -18021.0, AR(0.55) = [0.51], Max-Change = 0.0010
Stage 1 = 43, CDLL = -18007.9, AR(0.55) = [0.53], Max-Change = 0.0062
Stage 1 = 44, CDLL = -18055.7, AR(0.55) = [0.53], Max-Change = 0.0064
Stage 1 = 45, CDLL = -18007.2, AR(0.55) = [0.49], Max-Change = 0.0153
Stage 1 = 46, CDLL = -18038.6, AR(0.55) = [0.54], Max-Change = 0.0054
Stage 1 = 47, CDLL = -18023.6, AR(0.55) = [0.53], Max-Change = 0.0047
Stage 1 = 48, CDLL = -18058.8, AR(0.55) = [0.49], Max-Change = 0.0053
Stage 1 = 49, CDLL = -18019.5, AR(0.55) = [0.51], Max-Change = 0.0045
Stage 1 = 50, CDLL = -18044.3, AR(0.55) = [0.51], Max-Change = 0.0022
Stage 1 = 51, CDLL = -18083.4, AR(0.55) = [0.53], Max-Change = 0.0027
Stage 1 = 52, CDLL = -18051.1, AR(0.55) = [0.49], Max-Change = 0.0027
Stage 1 = 53, CDLL = -18055.0, AR(0.55) = [0.51], Max-Change = 0.0048
Stage 1 = 54, CDLL = -18010.1, AR(0.55) = [0.54], Max-Change = 0.0051
Stage 1 = 55, CDLL = -18061.8, AR(0.55) = [0.53], Max-Change = 0.0057
Stage 1 = 56, CDLL = -18044.9, AR(0.55) = [0.53], Max-Change = 0.0030
Stage 1 = 57, CDLL = -18049.0, AR(0.55) = [0.50], Max-Change = 0.0083
Stage 1 = 58, CDLL = -17988.5, AR(0.55) = [0.54], Max-Change = 0.0047
Stage 1 = 59, CDLL = -18027.6, AR(0.55) = [0.49], Max-Change = 0.0068
Stage 1 = 60, CDLL = -18063.1, AR(0.55) = [0.51], Max-Change = 0.0042
Stage 1 = 61, CDLL = -18081.5, AR(0.55) = [0.52], Max-Change = 0.0099
Stage 1 = 62, CDLL = -18025.8, AR(0.55) = [0.51], Max-Change = 0.0008
Stage 1 = 63, CDLL = -18039.0, AR(0.55) = [0.51], Max-Change = 0.0020
Stage 1 = 64, CDLL = -18024.9, AR(0.55) = [0.53], Max-Change = 0.0040
Stage 1 = 65, CDLL = -18038.7, AR(0.55) = [0.51], Max-Change = 0.0047
Stage 1 = 66, CDLL = -18030.0, AR(0.55) = [0.50], Max-Change = 0.0017
Stage 1 = 67, CDLL = -18068.9, AR(0.55) = [0.49], Max-Change = 0.0028
Stage 1 = 68, CDLL = -18051.2, AR(0.55) = [0.53], Max-Change = 0.0052
Stage 1 = 69, CDLL = -18041.8, AR(0.55) = [0.54], Max-Change = 0.0014
Stage 1 = 70, CDLL = -18037.5, AR(0.55) = [0.51], Max-Change = 0.0060
Stage 1 = 71, CDLL = -18026.9, AR(0.55) = [0.51], Max-Change = 0.0037
Stage 1 = 72, CDLL = -18049.5, AR(0.55) = [0.52], Max-Change = 0.0053
Stage 1 = 73, CDLL = -18042.4, AR(0.55) = [0.54], Max-Change = 0.0065
Stage 1 = 74, CDLL = -17998.7, AR(0.55) = [0.52], Max-Change = 0.0096
Stage 1 = 75, CDLL = -18046.2, AR(0.55) = [0.49], Max-Change = 0.0029
Stage 1 = 76, CDLL = -18039.9, AR(0.55) = [0.48], Max-Change = 0.0065
Stage 1 = 77, CDLL = -18058.5, AR(0.55) = [0.52], Max-Change = 0.0028
Stage 1 = 78, CDLL = -18055.3, AR(0.55) = [0.49], Max-Change = 0.0079
Stage 1 = 79, CDLL = -18049.3, AR(0.55) = [0.54], Max-Change = 0.0044
Stage 1 = 80, CDLL = -18011.4, AR(0.55) = [0.50], Max-Change = 0.0039
Stage 1 = 81, CDLL = -18051.2, AR(0.55) = [0.50], Max-Change = 0.0013
Stage 1 = 82, CDLL = -18003.3, AR(0.55) = [0.52], Max-Change = 0.0054
Stage 1 = 83, CDLL = -18012.3, AR(0.55) = [0.51], Max-Change = 0.0084
Stage 1 = 84, CDLL = -18036.1, AR(0.55) = [0.52], Max-Change = 0.0063
Stage 1 = 85, CDLL = -17987.1, AR(0.55) = [0.49], Max-Change = 0.0108
Stage 1 = 86, CDLL = -17996.7, AR(0.55) = [0.50], Max-Change = 0.0069
Stage 1 = 87, CDLL = -18020.4, AR(0.55) = [0.55], Max-Change = 0.0043
Stage 1 = 88, CDLL = -18047.0, AR(0.55) = [0.52], Max-Change = 0.0076
Stage 1 = 89, CDLL = -18023.2, AR(0.55) = [0.55], Max-Change = 0.0081
Stage 1 = 90, CDLL = -18035.7, AR(0.55) = [0.51], Max-Change = 0.0039
Stage 1 = 91, CDLL = -18043.6, AR(0.55) = [0.53], Max-Change = 0.0033
Stage 1 = 92, CDLL = -18017.3, AR(0.55) = [0.51], Max-Change = 0.0061
Stage 1 = 93, CDLL = -18000.6, AR(0.55) = [0.49], Max-Change = 0.0015
Stage 1 = 94, CDLL = -18021.8, AR(0.55) = [0.52], Max-Change = 0.0011
Stage 1 = 95, CDLL = -18048.8, AR(0.55) = [0.52], Max-Change = 0.0038
Stage 1 = 96, CDLL = -18023.6, AR(0.55) = [0.53], Max-Change = 0.0028
Stage 1 = 97, CDLL = -18066.4, AR(0.55) = [0.51], Max-Change = 0.0071
Stage 1 = 98, CDLL = -18029.1, AR(0.55) = [0.51], Max-Change = 0.0045
Stage 1 = 99, CDLL = -18058.2, AR(0.55) = [0.47], Max-Change = 0.0088
Stage 1 = 100, CDLL = -18020.8, AR(0.55) = [0.50], Max-Change = 0.0041
Stage 1 = 101, CDLL = -18029.9, AR(0.55) = [0.49], Max-Change = 0.0047
Stage 1 = 102, CDLL = -18040.5, AR(0.55) = [0.52], Max-Change = 0.0066
Stage 1 = 103, CDLL = -18043.0, AR(0.55) = [0.52], Max-Change = 0.0012
Stage 1 = 104, CDLL = -18020.7, AR(0.55) = [0.49], Max-Change = 0.0042
Stage 1 = 105, CDLL = -18015.0, AR(0.55) = [0.50], Max-Change = 0.0047
Stage 1 = 106, CDLL = -18018.4, AR(0.55) = [0.49], Max-Change = 0.0012
Stage 1 = 107, CDLL = -18047.9, AR(0.55) = [0.55], Max-Change = 0.0045
Stage 1 = 108, CDLL = -18020.7, AR(0.55) = [0.53], Max-Change = 0.0100
Stage 1 = 109, CDLL = -18028.9, AR(0.55) = [0.52], Max-Change = 0.0054
Stage 1 = 110, CDLL = -18048.6, AR(0.55) = [0.50], Max-Change = 0.0042
Stage 1 = 111, CDLL = -18023.1, AR(0.55) = [0.51], Max-Change = 0.0077
Stage 1 = 112, CDLL = -18055.0, AR(0.55) = [0.51], Max-Change = 0.0021
Stage 1 = 113, CDLL = -17997.7, AR(0.55) = [0.53], Max-Change = 0.0048
Stage 1 = 114, CDLL = -18053.8, AR(0.55) = [0.52], Max-Change = 0.0027
Stage 1 = 115, CDLL = -18065.2, AR(0.55) = [0.50], Max-Change = 0.0073
Stage 1 = 116, CDLL = -18016.5, AR(0.55) = [0.52], Max-Change = 0.0024
Stage 1 = 117, CDLL = -18048.6, AR(0.55) = [0.53], Max-Change = 0.0015
Stage 1 = 118, CDLL = -18086.5, AR(0.55) = [0.51], Max-Change = 0.0032
Stage 1 = 119, CDLL = -18043.2, AR(0.55) = [0.51], Max-Change = 0.0027
Stage 1 = 120, CDLL = -18023.4, AR(0.55) = [0.51], Max-Change = 0.0120
Stage 1 = 121, CDLL = -18014.3, AR(0.55) = [0.54], Max-Change = 0.0035
Stage 1 = 122, CDLL = -18014.7, AR(0.55) = [0.51], Max-Change = 0.0022
Stage 1 = 123, CDLL = -18028.3, AR(0.55) = [0.48], Max-Change = 0.0106
Stage 1 = 124, CDLL = -17993.6, AR(0.55) = [0.51], Max-Change = 0.0044
Stage 1 = 125, CDLL = -18029.1, AR(0.55) = [0.53], Max-Change = 0.0058
Stage 1 = 126, CDLL = -18056.6, AR(0.55) = [0.48], Max-Change = 0.0075
Stage 1 = 127, CDLL = -18029.6, AR(0.55) = [0.54], Max-Change = 0.0028
Stage 1 = 128, CDLL = -17986.4, AR(0.55) = [0.50], Max-Change = 0.0017
Stage 1 = 129, CDLL = -17980.8, AR(0.55) = [0.54], Max-Change = 0.0069
Stage 1 = 130, CDLL = -18005.1, AR(0.55) = [0.51], Max-Change = 0.0057
Stage 1 = 131, CDLL = -18014.1, AR(0.55) = [0.55], Max-Change = 0.0056
Stage 1 = 132, CDLL = -18032.3, AR(0.55) = [0.49], Max-Change = 0.0048
Stage 1 = 133, CDLL = -17995.5, AR(0.55) = [0.54], Max-Change = 0.0031
Stage 1 = 134, CDLL = -18007.4, AR(0.55) = [0.51], Max-Change = 0.0047
Stage 1 = 135, CDLL = -18029.5, AR(0.55) = [0.52], Max-Change = 0.0034
Stage 1 = 136, CDLL = -18037.1, AR(0.55) = [0.50], Max-Change = 0.0017
Stage 1 = 137, CDLL = -18033.0, AR(0.55) = [0.49], Max-Change = 0.0042
Stage 1 = 138, CDLL = -18013.3, AR(0.55) = [0.50], Max-Change = 0.0047
Stage 1 = 139, CDLL = -18028.3, AR(0.55) = [0.51], Max-Change = 0.0013
Stage 1 = 140, CDLL = -18068.8, AR(0.55) = [0.50], Max-Change = 0.0039
Stage 1 = 141, CDLL = -18092.7, AR(0.55) = [0.53], Max-Change = 0.0122
Stage 1 = 142, CDLL = -18027.1, AR(0.55) = [0.52], Max-Change = 0.0016
Stage 1 = 143, CDLL = -18034.3, AR(0.55) = [0.51], Max-Change = 0.0027
Stage 1 = 144, CDLL = -18047.1, AR(0.55) = [0.53], Max-Change = 0.0057
Stage 1 = 145, CDLL = -18060.0, AR(0.55) = [0.51], Max-Change = 0.0020
Stage 1 = 146, CDLL = -18011.2, AR(0.55) = [0.50], Max-Change = 0.0035
Stage 1 = 147, CDLL = -18018.3, AR(0.55) = [0.52], Max-Change = 0.0056
Stage 1 = 148, CDLL = -18041.0, AR(0.55) = [0.51], Max-Change = 0.0021
Stage 1 = 149, CDLL = -18067.7, AR(0.55) = [0.52], Max-Change = 0.0062
Stage 1 = 150, CDLL = -18074.5, AR(1.48) = [0.37], Max-Change = 0.0043
Stage 2 = 1, CDLL = -18075.0, AR(1.48) = [0.36], Max-Change = 0.0035
Stage 2 = 2, CDLL = -18020.4, AR(1.48) = [0.37], Max-Change = 0.0027
Stage 2 = 3, CDLL = -18036.7, AR(1.48) = [0.35], Max-Change = 0.0062
Stage 2 = 4, CDLL = -18007.8, AR(1.48) = [0.37], Max-Change = 0.0089
Stage 2 = 5, CDLL = -18050.5, AR(1.48) = [0.35], Max-Change = 0.0049
Stage 2 = 6, CDLL = -18028.9, AR(1.48) = [0.33], Max-Change = 0.0021
Stage 2 = 7, CDLL = -18030.7, AR(1.48) = [0.36], Max-Change = 0.0077
Stage 2 = 8, CDLL = -18029.3, AR(1.48) = [0.37], Max-Change = 0.0072
Stage 2 = 9, CDLL = -18039.6, AR(1.48) = [0.35], Max-Change = 0.0049
Stage 2 = 10, CDLL = -18045.8, AR(1.48) = [0.39], Max-Change = 0.0053
Stage 2 = 11, CDLL = -18031.1, AR(1.48) = [0.36], Max-Change = 0.0020
Stage 2 = 12, CDLL = -18014.4, AR(1.48) = [0.35], Max-Change = 0.0138
Stage 2 = 13, CDLL = -18041.6, AR(1.48) = [0.35], Max-Change = 0.0036
Stage 2 = 14, CDLL = -18057.4, AR(1.48) = [0.35], Max-Change = 0.0050
Stage 2 = 15, CDLL = -18039.7, AR(1.48) = [0.37], Max-Change = 0.0045
Stage 2 = 16, CDLL = -18026.6, AR(1.48) = [0.37], Max-Change = 0.0031
Stage 2 = 17, CDLL = -18046.9, AR(1.48) = [0.34], Max-Change = 0.0139
Stage 2 = 18, CDLL = -18066.6, AR(1.48) = [0.37], Max-Change = 0.0050
Stage 2 = 19, CDLL = -18019.5, AR(1.48) = [0.36], Max-Change = 0.0058
Stage 2 = 20, CDLL = -18042.3, AR(1.48) = [0.36], Max-Change = 0.0020
Stage 2 = 21, CDLL = -18037.4, AR(1.48) = [0.37], Max-Change = 0.0034
Stage 2 = 22, CDLL = -18045.2, AR(1.48) = [0.34], Max-Change = 0.0048
Stage 2 = 23, CDLL = -18067.0, AR(1.48) = [0.36], Max-Change = 0.0042
Stage 2 = 24, CDLL = -18025.6, AR(1.48) = [0.34], Max-Change = 0.0048
Stage 2 = 25, CDLL = -18019.6, AR(1.48) = [0.39], Max-Change = 0.0025
Stage 2 = 26, CDLL = -18031.9, AR(1.48) = [0.35], Max-Change = 0.0083
Stage 2 = 27, CDLL = -18034.4, AR(1.48) = [0.37], Max-Change = 0.0053
Stage 2 = 28, CDLL = -18012.9, AR(1.48) = [0.36], Max-Change = 0.0025
Stage 2 = 29, CDLL = -18040.2, AR(1.48) = [0.35], Max-Change = 0.0007
Stage 2 = 30, CDLL = -18013.3, AR(1.48) = [0.38], Max-Change = 0.0064
Stage 2 = 31, CDLL = -18023.0, AR(1.48) = [0.33], Max-Change = 0.0035
Stage 2 = 32, CDLL = -17992.1, AR(1.48) = [0.37], Max-Change = 0.0074
Stage 2 = 33, CDLL = -18050.9, AR(1.48) = [0.35], Max-Change = 0.0026
Stage 2 = 34, CDLL = -18036.3, AR(1.48) = [0.33], Max-Change = 0.0026
Stage 2 = 35, CDLL = -18038.8, AR(1.48) = [0.35], Max-Change = 0.0031
Stage 2 = 36, CDLL = -18063.6, AR(1.48) = [0.36], Max-Change = 0.0013
Stage 2 = 37, CDLL = -18034.2, AR(1.48) = [0.38], Max-Change = 0.0035
Stage 2 = 38, CDLL = -18072.4, AR(1.48) = [0.39], Max-Change = 0.0077
Stage 2 = 39, CDLL = -18067.9, AR(1.48) = [0.39], Max-Change = 0.0029
Stage 2 = 40, CDLL = -18066.4, AR(1.48) = [0.36], Max-Change = 0.0078
Stage 2 = 41, CDLL = -17994.7, AR(1.48) = [0.37], Max-Change = 0.0004
Stage 2 = 42, CDLL = -18015.2, AR(1.48) = [0.37], Max-Change = 0.0022
Stage 2 = 43, CDLL = -18020.0, AR(1.48) = [0.35], Max-Change = 0.0019
Stage 2 = 44, CDLL = -18042.6, AR(1.48) = [0.35], Max-Change = 0.0074
Stage 2 = 45, CDLL = -18028.3, AR(1.48) = [0.36], Max-Change = 0.0029
Stage 2 = 46, CDLL = -18045.4, AR(1.48) = [0.36], Max-Change = 0.0012
Stage 2 = 47, CDLL = -18043.9, AR(1.48) = [0.35], Max-Change = 0.0032
Stage 2 = 48, CDLL = -18030.7, AR(1.48) = [0.35], Max-Change = 0.0078
Stage 2 = 49, CDLL = -18074.4, AR(1.48) = [0.38], Max-Change = 0.0016
Stage 2 = 50, CDLL = -18029.5, AR(1.48) = [0.36], Max-Change = 0.0036
Stage 2 = 51, CDLL = -18045.6, AR(1.48) = [0.35], Max-Change = 0.0012
Stage 2 = 52, CDLL = -18011.5, AR(1.48) = [0.38], Max-Change = 0.0019
Stage 2 = 53, CDLL = -18049.7, AR(1.48) = [0.37], Max-Change = 0.0072
Stage 2 = 54, CDLL = -18058.8, AR(1.48) = [0.35], Max-Change = 0.0032
Stage 2 = 55, CDLL = -18050.6, AR(1.48) = [0.36], Max-Change = 0.0033
Stage 2 = 56, CDLL = -18038.6, AR(1.48) = [0.34], Max-Change = 0.0015
Stage 2 = 57, CDLL = -18061.3, AR(1.48) = [0.35], Max-Change = 0.0042
Stage 2 = 58, CDLL = -18094.5, AR(1.48) = [0.37], Max-Change = 0.0065
Stage 2 = 59, CDLL = -18066.0, AR(1.48) = [0.36], Max-Change = 0.0056
Stage 2 = 60, CDLL = -18046.5, AR(1.48) = [0.37], Max-Change = 0.0046
Stage 2 = 61, CDLL = -18079.2, AR(1.48) = [0.39], Max-Change = 0.0031
Stage 2 = 62, CDLL = -18056.8, AR(1.48) = [0.37], Max-Change = 0.0111
Stage 2 = 63, CDLL = -18055.0, AR(1.48) = [0.37], Max-Change = 0.0063
Stage 2 = 64, CDLL = -18068.9, AR(1.48) = [0.35], Max-Change = 0.0039
Stage 2 = 65, CDLL = -18003.1, AR(1.48) = [0.37], Max-Change = 0.0057
Stage 2 = 66, CDLL = -18035.7, AR(1.48) = [0.35], Max-Change = 0.0079
Stage 2 = 67, CDLL = -18003.6, AR(1.48) = [0.37], Max-Change = 0.0120
Stage 2 = 68, CDLL = -18048.1, AR(1.48) = [0.37], Max-Change = 0.0070
Stage 2 = 69, CDLL = -18021.0, AR(1.48) = [0.41], Max-Change = 0.0010
Stage 2 = 70, CDLL = -18020.0, AR(1.48) = [0.37], Max-Change = 0.0039
Stage 2 = 71, CDLL = -18015.4, AR(1.48) = [0.34], Max-Change = 0.0052
Stage 2 = 72, CDLL = -18003.6, AR(1.48) = [0.39], Max-Change = 0.0026
Stage 2 = 73, CDLL = -18055.3, AR(1.48) = [0.36], Max-Change = 0.0030
Stage 2 = 74, CDLL = -18032.3, AR(1.48) = [0.38], Max-Change = 0.0034
Stage 2 = 75, CDLL = -18045.2, AR(1.48) = [0.36], Max-Change = 0.0034
Stage 2 = 76, CDLL = -18031.2, AR(1.48) = [0.31], Max-Change = 0.0028
Stage 2 = 77, CDLL = -18016.4, AR(1.48) = [0.38], Max-Change = 0.0040
Stage 2 = 78, CDLL = -18008.2, AR(1.48) = [0.36], Max-Change = 0.0089
Stage 2 = 79, CDLL = -18042.4, AR(1.48) = [0.37], Max-Change = 0.0048
Stage 2 = 80, CDLL = -18002.3, AR(1.48) = [0.35], Max-Change = 0.0042
Stage 2 = 81, CDLL = -18029.3, AR(1.48) = [0.35], Max-Change = 0.0025
Stage 2 = 82, CDLL = -18038.5, AR(1.48) = [0.36], Max-Change = 0.0034
Stage 2 = 83, CDLL = -18054.1, AR(1.48) = [0.39], Max-Change = 0.0053
Stage 2 = 84, CDLL = -17992.5, AR(1.48) = [0.35], Max-Change = 0.0042
Stage 2 = 85, CDLL = -17976.0, AR(1.48) = [0.37], Max-Change = 0.0016
Stage 2 = 86, CDLL = -18027.2, AR(1.48) = [0.36], Max-Change = 0.0026
Stage 2 = 87, CDLL = -18022.9, AR(1.48) = [0.35], Max-Change = 0.0043
Stage 2 = 88, CDLL = -17998.7, AR(1.48) = [0.35], Max-Change = 0.0010
Stage 2 = 89, CDLL = -18011.3, AR(1.48) = [0.35], Max-Change = 0.0039
Stage 2 = 90, CDLL = -17993.7, AR(1.48) = [0.34], Max-Change = 0.0053
Stage 2 = 91, CDLL = -18023.8, AR(1.48) = [0.35], Max-Change = 0.0094
Stage 2 = 92, CDLL = -18029.0, AR(1.48) = [0.37], Max-Change = 0.0066
Stage 2 = 93, CDLL = -18053.3, AR(1.48) = [0.39], Max-Change = 0.0042
Stage 2 = 94, CDLL = -18038.0, AR(1.48) = [0.35], Max-Change = 0.0055
Stage 2 = 95, CDLL = -18021.7, AR(1.48) = [0.34], Max-Change = 0.0049
Stage 2 = 96, CDLL = -18008.2, AR(1.48) = [0.36], Max-Change = 0.0048
Stage 2 = 97, CDLL = -18057.6, AR(1.48) = [0.36], Max-Change = 0.0028
Stage 2 = 98, CDLL = -18050.3, AR(1.48) = [0.34], Max-Change = 0.0043
Stage 2 = 99, CDLL = -18041.1, AR(1.48) = [0.38], Max-Change = 0.0065
Stage 2 = 100, CDLL = -18005.8, AR(1.48) = [0.36], Max-Change = 0.0139
Stage 3 = 1, CDLL = -18051.5, AR(1.48) = [0.38], gam = 0.0000, Max-Change = 0.0000
Stage 3 = 2, CDLL = -18068.6, AR(1.48) = [0.37], gam = 0.1778, Max-Change = 0.0035
Stage 3 = 3, CDLL = -18034.2, AR(1.48) = [0.37], gam = 0.1057, Max-Change = 0.0014
Stage 3 = 4, CDLL = -17995.7, AR(1.48) = [0.37], gam = 0.0780, Max-Change = 0.0018
Stage 3 = 5, CDLL = -18021.6, AR(1.48) = [0.39], gam = 0.0629, Max-Change = 0.0010
Stage 3 = 6, CDLL = -18043.9, AR(1.48) = [0.38], gam = 0.0532, Max-Change = 0.0012
Stage 3 = 7, CDLL = -18043.4, AR(1.48) = [0.36], gam = 0.0464, Max-Change = 0.0004
Stage 3 = 8, CDLL = -18037.1, AR(1.48) = [0.38], gam = 0.0413, Max-Change = 0.0010
Stage 3 = 9, CDLL = -18046.3, AR(1.48) = [0.38], gam = 0.0374, Max-Change = 0.0007

#> 
#> Calculating log-likelihood...
summary(LLTM)
#> 
#> Call:
#> mixedmirt(data = dat, model = 1, fixed = ~0 + difficulty, itemdesign = itemdesign, 
#>     SE = FALSE)
#> 
#> --------------
#> FIXED EFFECTS:
#>                  Estimate Std.Error z.value
#> difficultyeasy      1.113        NA      NA
#> difficultyhard     -0.933        NA      NA
#> difficultymedium    0.054        NA      NA
#> 
#> --------------
#> RANDOM EFFECT COVARIANCE(S):
#> Correlations on upper diagonal
#> 
#> $Theta
#>       F1
#> F1 0.957
#> 
coef(LLTM)
#> $Item_1
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_2
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_3
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_4
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_5
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_6
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_7
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_8
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_9
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_10
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_11
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_12
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_13
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_14
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_15
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_16
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_17
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_18
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_19
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_20
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_21
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_22
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_23
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_24
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_25
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_26
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_27
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_28
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_29
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $Item_30
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.113         -0.933            0.054  1 0 0 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0  0.957
#> 

# LLTM with random error estimate (not supported with mirt() )
LLTM.e <- mixedmirt(dat, model=1, fixed = ~ 0 + difficulty,
                  random = ~ 1|items, itemdesign=itemdesign, SE=FALSE)
#> 
Stage 1 = 1, CDLL = -19926.5, AR(0.55) = [0.51], Max-Change = 0.1895
Stage 1 = 2, CDLL = -19273.2, AR(0.55) = [0.51], Max-Change = 0.1532
Stage 1 = 3, CDLL = -18887.8, AR(0.55) = [0.50], Max-Change = 0.1314
Stage 1 = 4, CDLL = -18569.4, AR(0.55) = [0.47], Max-Change = 0.1007
Stage 1 = 5, CDLL = -18373.7, AR(0.55) = [0.48], Max-Change = 0.0870
Stage 1 = 6, CDLL = -18243.0, AR(0.55) = [0.48], Max-Change = 0.0715
Stage 1 = 7, CDLL = -18109.2, AR(0.55) = [0.49], Max-Change = 0.0569
Stage 1 = 8, CDLL = -18135.3, AR(0.55) = [0.53], Max-Change = 0.0476
Stage 1 = 9, CDLL = -18055.8, AR(0.55) = [0.53], Max-Change = 0.0393
Stage 1 = 10, CDLL = -18031.8, AR(0.55) = [0.50], Max-Change = 0.0356
Stage 1 = 11, CDLL = -18010.3, AR(0.55) = [0.52], Max-Change = 0.0282
Stage 1 = 12, CDLL = -18023.4, AR(0.55) = [0.48], Max-Change = 0.0228
Stage 1 = 13, CDLL = -18031.6, AR(0.55) = [0.53], Max-Change = 0.0195
Stage 1 = 14, CDLL = -18037.1, AR(0.55) = [0.49], Max-Change = 0.0186
Stage 1 = 15, CDLL = -18042.3, AR(0.55) = [0.51], Max-Change = 0.0167
Stage 1 = 16, CDLL = -18021.7, AR(0.55) = [0.51], Max-Change = 0.0112
Stage 1 = 17, CDLL = -18015.7, AR(0.55) = [0.51], Max-Change = 0.0157
Stage 1 = 18, CDLL = -18040.2, AR(0.55) = [0.52], Max-Change = 0.0056
Stage 1 = 19, CDLL = -18055.3, AR(0.55) = [0.49], Max-Change = 0.0061
Stage 1 = 20, CDLL = -18031.5, AR(0.55) = [0.50], Max-Change = 0.0057
Stage 1 = 21, CDLL = -18005.0, AR(0.55) = [0.49], Max-Change = 0.0061
Stage 1 = 22, CDLL = -18014.5, AR(0.55) = [0.51], Max-Change = 0.0064
Stage 1 = 23, CDLL = -18035.0, AR(0.55) = [0.53], Max-Change = 0.0048
Stage 1 = 24, CDLL = -18050.3, AR(0.55) = [0.52], Max-Change = 0.0053
Stage 1 = 25, CDLL = -17996.4, AR(0.55) = [0.50], Max-Change = 0.0050
Stage 1 = 26, CDLL = -18021.3, AR(0.55) = [0.51], Max-Change = 0.0128
Stage 1 = 27, CDLL = -18074.3, AR(0.55) = [0.55], Max-Change = 0.0070
Stage 1 = 28, CDLL = -18016.5, AR(0.55) = [0.52], Max-Change = 0.0037
Stage 1 = 29, CDLL = -18022.5, AR(0.55) = [0.49], Max-Change = 0.0043
Stage 1 = 30, CDLL = -18031.9, AR(0.55) = [0.53], Max-Change = 0.0089
Stage 1 = 31, CDLL = -18024.6, AR(0.55) = [0.51], Max-Change = 0.0025
Stage 1 = 32, CDLL = -18024.2, AR(0.55) = [0.51], Max-Change = 0.0037
Stage 1 = 33, CDLL = -17984.2, AR(0.55) = [0.52], Max-Change = 0.0030
Stage 1 = 34, CDLL = -18003.4, AR(0.55) = [0.50], Max-Change = 0.0023
Stage 1 = 35, CDLL = -18014.3, AR(0.55) = [0.51], Max-Change = 0.0061
Stage 1 = 36, CDLL = -18046.5, AR(0.55) = [0.53], Max-Change = 0.0037
Stage 1 = 37, CDLL = -18019.2, AR(0.55) = [0.54], Max-Change = 0.0037
Stage 1 = 38, CDLL = -18020.1, AR(0.55) = [0.53], Max-Change = 0.0008
Stage 1 = 39, CDLL = -18072.9, AR(0.55) = [0.52], Max-Change = 0.0058
Stage 1 = 40, CDLL = -18016.6, AR(0.55) = [0.51], Max-Change = 0.0078
Stage 1 = 41, CDLL = -18050.3, AR(0.55) = [0.52], Max-Change = 0.0026
Stage 1 = 42, CDLL = -18021.0, AR(0.55) = [0.51], Max-Change = 0.0010
Stage 1 = 43, CDLL = -18007.9, AR(0.55) = [0.53], Max-Change = 0.0062
Stage 1 = 44, CDLL = -18055.7, AR(0.55) = [0.53], Max-Change = 0.0064
Stage 1 = 45, CDLL = -18007.2, AR(0.55) = [0.49], Max-Change = 0.0153
Stage 1 = 46, CDLL = -18038.6, AR(0.55) = [0.54], Max-Change = 0.0054
Stage 1 = 47, CDLL = -18023.6, AR(0.55) = [0.53], Max-Change = 0.0047
Stage 1 = 48, CDLL = -18058.8, AR(0.55) = [0.49], Max-Change = 0.0053
Stage 1 = 49, CDLL = -18019.5, AR(0.55) = [0.51], Max-Change = 0.0045
Stage 1 = 50, CDLL = -18044.3, AR(0.55) = [0.51], Max-Change = 0.0022
Stage 1 = 51, CDLL = -18083.4, AR(0.55) = [0.53], Max-Change = 0.0027
Stage 1 = 52, CDLL = -18051.1, AR(0.55) = [0.49], Max-Change = 0.0027
Stage 1 = 53, CDLL = -18055.0, AR(0.55) = [0.51], Max-Change = 0.0048
Stage 1 = 54, CDLL = -18010.1, AR(0.55) = [0.54], Max-Change = 0.0051
Stage 1 = 55, CDLL = -18061.8, AR(0.55) = [0.53], Max-Change = 0.0057
Stage 1 = 56, CDLL = -18044.9, AR(0.55) = [0.53], Max-Change = 0.0030
Stage 1 = 57, CDLL = -18049.0, AR(0.55) = [0.50], Max-Change = 0.0083
Stage 1 = 58, CDLL = -17988.5, AR(0.55) = [0.54], Max-Change = 0.0047
Stage 1 = 59, CDLL = -18027.6, AR(0.55) = [0.49], Max-Change = 0.0068
Stage 1 = 60, CDLL = -18063.1, AR(0.55) = [0.51], Max-Change = 0.0042
Stage 1 = 61, CDLL = -18081.5, AR(0.55) = [0.52], Max-Change = 0.0099
Stage 1 = 62, CDLL = -18025.8, AR(0.55) = [0.51], Max-Change = 0.0008
Stage 1 = 63, CDLL = -18039.0, AR(0.55) = [0.51], Max-Change = 0.0020
Stage 1 = 64, CDLL = -18024.9, AR(0.55) = [0.53], Max-Change = 0.0040
Stage 1 = 65, CDLL = -18038.7, AR(0.55) = [0.51], Max-Change = 0.0047
Stage 1 = 66, CDLL = -18030.0, AR(0.55) = [0.50], Max-Change = 0.0017
Stage 1 = 67, CDLL = -18068.9, AR(0.55) = [0.49], Max-Change = 0.0028
Stage 1 = 68, CDLL = -18051.2, AR(0.55) = [0.53], Max-Change = 0.0052
Stage 1 = 69, CDLL = -18041.8, AR(0.55) = [0.54], Max-Change = 0.0014
Stage 1 = 70, CDLL = -18037.5, AR(0.55) = [0.51], Max-Change = 0.0060
Stage 1 = 71, CDLL = -18026.9, AR(0.55) = [0.51], Max-Change = 0.0037
Stage 1 = 72, CDLL = -18049.5, AR(0.55) = [0.52], Max-Change = 0.0053
Stage 1 = 73, CDLL = -18042.4, AR(0.55) = [0.54], Max-Change = 0.0065
Stage 1 = 74, CDLL = -17998.7, AR(0.55) = [0.52], Max-Change = 0.0096
Stage 1 = 75, CDLL = -18046.2, AR(0.55) = [0.49], Max-Change = 0.0029
Stage 1 = 76, CDLL = -18039.9, AR(0.55) = [0.48], Max-Change = 0.0065
Stage 1 = 77, CDLL = -18058.5, AR(0.55) = [0.52], Max-Change = 0.0028
Stage 1 = 78, CDLL = -18055.3, AR(0.55) = [0.49], Max-Change = 0.0079
Stage 1 = 79, CDLL = -18049.3, AR(0.55) = [0.54], Max-Change = 0.0044
Stage 1 = 80, CDLL = -18011.4, AR(0.55) = [0.50], Max-Change = 0.0039
Stage 1 = 81, CDLL = -18051.2, AR(0.55) = [0.50], Max-Change = 0.0013
Stage 1 = 82, CDLL = -18003.3, AR(0.55) = [0.52], Max-Change = 0.0054
Stage 1 = 83, CDLL = -18012.3, AR(0.55) = [0.51], Max-Change = 0.0084
Stage 1 = 84, CDLL = -18036.1, AR(0.55) = [0.52], Max-Change = 0.0063
Stage 1 = 85, CDLL = -17987.1, AR(0.55) = [0.49], Max-Change = 0.0108
Stage 1 = 86, CDLL = -17996.7, AR(0.55) = [0.50], Max-Change = 0.0069
Stage 1 = 87, CDLL = -18020.4, AR(0.55) = [0.55], Max-Change = 0.0043
Stage 1 = 88, CDLL = -18047.0, AR(0.55) = [0.52], Max-Change = 0.0076
Stage 1 = 89, CDLL = -18023.2, AR(0.55) = [0.55], Max-Change = 0.0081
Stage 1 = 90, CDLL = -18035.7, AR(0.55) = [0.51], Max-Change = 0.0039
Stage 1 = 91, CDLL = -18043.6, AR(0.55) = [0.53], Max-Change = 0.0033
Stage 1 = 92, CDLL = -18017.3, AR(0.55) = [0.51], Max-Change = 0.0061
Stage 1 = 93, CDLL = -18000.6, AR(0.55) = [0.49], Max-Change = 0.0015
Stage 1 = 94, CDLL = -18021.8, AR(0.55) = [0.52], Max-Change = 0.0011
Stage 1 = 95, CDLL = -18048.8, AR(0.55) = [0.52], Max-Change = 0.0038
Stage 1 = 96, CDLL = -18023.6, AR(0.55) = [0.53], Max-Change = 0.0028
Stage 1 = 97, CDLL = -18066.4, AR(0.55) = [0.51], Max-Change = 0.0071
Stage 1 = 98, CDLL = -18029.1, AR(0.55) = [0.51], Max-Change = 0.0045
Stage 1 = 99, CDLL = -18058.2, AR(0.55) = [0.47], Max-Change = 0.0088
Stage 1 = 100, CDLL = -18051.9, AR(0.58) = [0.51], Max-Change = 0.0341
Stage 1 = 101, CDLL = -18032.9, AR(0.58; 0.01) = [0.47; 0.67], Max-Change = 0.2000
Stage 1 = 102, CDLL = -18046.8, AR(0.58; 0.01) = [0.50; 0.43], Max-Change = 0.1635
Stage 1 = 103, CDLL = -18032.9, AR(0.58; 0.01) = [0.52; 0.27], Max-Change = 0.1300
Stage 1 = 104, CDLL = -18032.7, AR(0.58; 0.01) = [0.52; 0.23], Max-Change = 0.1040
Stage 1 = 105, CDLL = -18032.8, AR(0.58; 0.01) = [0.50; 0.27], Max-Change = 0.0830
Stage 1 = 106, CDLL = -18034.3, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0663
Stage 1 = 107, CDLL = -18020.2, AR(0.58; 0.01) = [0.52; 0.10], Max-Change = 0.0533
Stage 1 = 108, CDLL = -18035.4, AR(0.58; 0.01) = [0.52; 0.13], Max-Change = 0.0424
Stage 1 = 109, CDLL = -18055.7, AR(0.58; 0.01) = [0.50; 0.10], Max-Change = 0.0339
Stage 1 = 110, CDLL = -18053.7, AR(0.58; 0.01) = [0.51; 0.10], Max-Change = 0.0282
Stage 1 = 111, CDLL = -18023.3, AR(0.58; 0.01) = [0.49; 0.17], Max-Change = 0.0229
Stage 1 = 112, CDLL = -18084.3, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0193
Stage 1 = 113, CDLL = -18032.6, AR(0.58; 0.01) = [0.54; 0.17], Max-Change = 0.0178
Stage 1 = 114, CDLL = -18025.5, AR(0.58; 0.01) = [0.52; 0.13], Max-Change = 0.1145
Stage 1 = 115, CDLL = -18031.8, AR(0.58; 0.01) = [0.48; 0.10], Max-Change = 0.0062
Stage 1 = 116, CDLL = -17993.0, AR(0.58; 0.01) = [0.51; 0.17], Max-Change = 0.0091
Stage 1 = 117, CDLL = -18051.9, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0025
Stage 1 = 118, CDLL = -18006.8, AR(0.58; 0.01) = [0.52; 0.17], Max-Change = 0.0052
Stage 1 = 119, CDLL = -18004.4, AR(0.58; 0.01) = [0.48; 0.20], Max-Change = 0.0141
Stage 1 = 120, CDLL = -18049.6, AR(0.58; 0.01) = [0.50; 0.07], Max-Change = 0.0049
Stage 1 = 121, CDLL = -18009.1, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0088
Stage 1 = 122, CDLL = -18026.9, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0028
Stage 1 = 123, CDLL = -18035.7, AR(0.58; 0.01) = [0.50; 0.03], Max-Change = 0.0065
Stage 1 = 124, CDLL = -18037.9, AR(0.58; 0.01) = [0.51; 0.03], Max-Change = 0.0036
Stage 1 = 125, CDLL = -18006.8, AR(0.58; 0.01) = [0.50; 0.10], Max-Change = 0.0063
Stage 1 = 126, CDLL = -18009.2, AR(0.58; 0.01) = [0.49; 0.07], Max-Change = 0.0043
Stage 1 = 127, CDLL = -18056.0, AR(0.58; 0.01) = [0.53; 0.03], Max-Change = 0.0107
Stage 1 = 128, CDLL = -18053.9, AR(0.58; 0.01) = [0.49; 0.00], Max-Change = 0.0078
Stage 1 = 129, CDLL = -18062.5, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0054
Stage 1 = 130, CDLL = -18055.2, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0060
Stage 1 = 131, CDLL = -18044.0, AR(0.58; 0.01) = [0.50; 0.00], Max-Change = 0.0064
Stage 1 = 132, CDLL = -18006.5, AR(0.58; 0.01) = [0.52; 0.10], Max-Change = 0.0061
Stage 1 = 133, CDLL = -18030.2, AR(0.58; 0.01) = [0.53; 0.03], Max-Change = 0.0041
Stage 1 = 134, CDLL = -17997.1, AR(0.58; 0.01) = [0.51; 0.00], Max-Change = 0.0079
Stage 1 = 135, CDLL = -18045.1, AR(0.58; 0.01) = [0.52; 0.00], Max-Change = 0.0029
Stage 1 = 136, CDLL = -18027.9, AR(0.58; 0.01) = [0.50; 0.00], Max-Change = 0.0062
Stage 1 = 137, CDLL = -18054.0, AR(0.58; 0.01) = [0.53; 0.03], Max-Change = 0.0038
Stage 1 = 138, CDLL = -18056.0, AR(0.58; 0.01) = [0.49; 0.03], Max-Change = 0.0057
Stage 1 = 139, CDLL = -17994.1, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0041
Stage 1 = 140, CDLL = -18006.2, AR(0.58; 0.01) = [0.48; 0.00], Max-Change = 0.0074
Stage 1 = 141, CDLL = -18026.0, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0043
Stage 1 = 142, CDLL = -18003.9, AR(0.58; 0.01) = [0.50; 0.03], Max-Change = 0.0042
Stage 1 = 143, CDLL = -18061.5, AR(0.58; 0.01) = [0.53; 0.00], Max-Change = 0.0062
Stage 1 = 144, CDLL = -18014.9, AR(0.58; 0.01) = [0.51; 0.03], Max-Change = 0.0068
Stage 1 = 145, CDLL = -17988.1, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0031
Stage 1 = 146, CDLL = -18025.2, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0048
Stage 1 = 147, CDLL = -18018.5, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0035
Stage 1 = 148, CDLL = -18036.7, AR(0.58; 0.01) = [0.50; 0.00], Max-Change = 0.0083
Stage 1 = 149, CDLL = -18064.7, AR(0.58; 0.01) = [0.54; 0.10], Max-Change = 0.0063
Stage 1 = 150, CDLL = -18050.6, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0028
Stage 1 = 151, CDLL = -18028.0, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0064
Stage 1 = 152, CDLL = -18031.2, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0028
Stage 1 = 153, CDLL = -18003.8, AR(0.58; 0.01) = [0.48; 0.07], Max-Change = 0.0016
Stage 1 = 154, CDLL = -18035.1, AR(0.58; 0.01) = [0.49; 0.03], Max-Change = 0.0061
Stage 1 = 155, CDLL = -18016.6, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0055
Stage 1 = 156, CDLL = -18023.3, AR(0.58; 0.01) = [0.50; 0.00], Max-Change = 0.0028
Stage 1 = 157, CDLL = -18027.4, AR(0.58; 0.01) = [0.51; 0.00], Max-Change = 0.0064
Stage 1 = 158, CDLL = -18002.2, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0035
Stage 1 = 159, CDLL = -18027.1, AR(0.58; 0.01) = [0.48; 0.07], Max-Change = 0.0055
Stage 1 = 160, CDLL = -18047.3, AR(0.58; 0.01) = [0.50; 0.07], Max-Change = 0.0029
Stage 1 = 161, CDLL = -18031.1, AR(0.58; 0.01) = [0.49; 0.07], Max-Change = 0.0067
Stage 1 = 162, CDLL = -18016.0, AR(0.58; 0.01) = [0.48; 0.00], Max-Change = 0.0079
Stage 1 = 163, CDLL = -18055.9, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0043
Stage 1 = 164, CDLL = -18027.9, AR(0.58; 0.01) = [0.49; 0.07], Max-Change = 0.0039
Stage 1 = 165, CDLL = -18052.5, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0079
Stage 1 = 166, CDLL = -18011.4, AR(0.58; 0.01) = [0.49; 0.03], Max-Change = 0.0066
Stage 1 = 167, CDLL = -18051.5, AR(0.58; 0.01) = [0.54; 0.03], Max-Change = 0.0066
Stage 1 = 168, CDLL = -17994.6, AR(0.58; 0.01) = [0.50; 0.03], Max-Change = 0.0023
Stage 1 = 169, CDLL = -18011.1, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0038
Stage 1 = 170, CDLL = -17993.1, AR(0.58; 0.01) = [0.48; 0.03], Max-Change = 0.0034
Stage 1 = 171, CDLL = -18058.6, AR(0.58; 0.01) = [0.50; 0.03], Max-Change = 0.0032
Stage 1 = 172, CDLL = -18015.1, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0030
Stage 1 = 173, CDLL = -18051.3, AR(0.58; 0.01) = [0.52; 0.00], Max-Change = 0.0055
Stage 1 = 174, CDLL = -18047.4, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0026
Stage 1 = 175, CDLL = -18053.0, AR(0.58; 0.01) = [0.51; 0.00], Max-Change = 0.0127
Stage 1 = 176, CDLL = -18039.1, AR(0.58; 0.01) = [0.51; 0.07], Max-Change = 0.0087
Stage 1 = 177, CDLL = -18023.6, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0037
Stage 1 = 178, CDLL = -18023.9, AR(0.58; 0.01) = [0.50; 0.00], Max-Change = 0.0246
Stage 1 = 179, CDLL = -18063.7, AR(0.58; 0.01) = [0.49; 0.07], Max-Change = 0.0031
Stage 1 = 180, CDLL = -18072.8, AR(0.58; 0.01) = [0.55; 0.03], Max-Change = 0.0068
Stage 1 = 181, CDLL = -18029.1, AR(0.58; 0.01) = [0.49; 0.03], Max-Change = 0.0036
Stage 1 = 182, CDLL = -18024.8, AR(0.58; 0.01) = [0.50; 0.07], Max-Change = 0.0135
Stage 1 = 183, CDLL = -18026.3, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0032
Stage 1 = 184, CDLL = -18038.0, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0025
Stage 1 = 185, CDLL = -18051.2, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0022
Stage 1 = 186, CDLL = -18073.2, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0055
Stage 1 = 187, CDLL = -18035.5, AR(0.58; 0.01) = [0.53; 0.00], Max-Change = 0.0048
Stage 1 = 188, CDLL = -18028.0, AR(0.58; 0.01) = [0.52; 0.03], Max-Change = 0.0045
Stage 1 = 189, CDLL = -18007.7, AR(0.58; 0.01) = [0.47; 0.07], Max-Change = 0.0106
Stage 1 = 190, CDLL = -18017.0, AR(0.58; 0.01) = [0.53; 0.00], Max-Change = 0.0054
Stage 1 = 191, CDLL = -18019.5, AR(0.58; 0.01) = [0.50; 0.10], Max-Change = 0.0101
Stage 1 = 192, CDLL = -18032.9, AR(0.58; 0.01) = [0.51; 0.03], Max-Change = 0.0064
Stage 1 = 193, CDLL = -18011.0, AR(0.58; 0.01) = [0.50; 0.10], Max-Change = 0.0050
Stage 1 = 194, CDLL = -18062.7, AR(0.58; 0.01) = [0.53; 0.07], Max-Change = 0.0030
Stage 1 = 195, CDLL = -18021.6, AR(0.58; 0.01) = [0.50; 0.07], Max-Change = 0.0055
Stage 1 = 196, CDLL = -17998.0, AR(0.58; 0.01) = [0.51; 0.00], Max-Change = 0.0140
Stage 1 = 197, CDLL = -18015.8, AR(0.58; 0.01) = [0.49; 0.03], Max-Change = 0.0070
Stage 1 = 198, CDLL = -18025.8, AR(0.58; 0.01) = [0.52; 0.07], Max-Change = 0.0040
Stage 1 = 199, CDLL = -18026.6, AR(0.58; 0.01) = [0.49; 0.13], Max-Change = 0.0064
Stage 1 = 200, CDLL = -18004.2, AR(1.16; 0.00) = [0.43; 0.17], Max-Change = 0.0030
Stage 2 = 1, CDLL = -17991.6, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0036
Stage 2 = 2, CDLL = -18049.4, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0038
Stage 2 = 3, CDLL = -18019.4, AR(1.16; 0.00) = [0.40; 0.07], Max-Change = 0.0027
Stage 2 = 4, CDLL = -18042.4, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0043
Stage 2 = 5, CDLL = -18032.7, AR(1.16; 0.00) = [0.40; 0.13], Max-Change = 0.0011
Stage 2 = 6, CDLL = -18014.1, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0067
Stage 2 = 7, CDLL = -18011.4, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0066
Stage 2 = 8, CDLL = -18041.4, AR(1.16; 0.00) = [0.42; 0.07], Max-Change = 0.0051
Stage 2 = 9, CDLL = -18061.8, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0036
Stage 2 = 10, CDLL = -18074.7, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0096
Stage 2 = 11, CDLL = -18051.2, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0032
Stage 2 = 12, CDLL = -18056.9, AR(1.16; 0.00) = [0.42; 0.07], Max-Change = 0.0064
Stage 2 = 13, CDLL = -18042.6, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0033
Stage 2 = 14, CDLL = -18050.6, AR(1.16; 0.00) = [0.37; 0.10], Max-Change = 0.0038
Stage 2 = 15, CDLL = -18024.6, AR(1.16; 0.00) = [0.41; 0.10], Max-Change = 0.0065
Stage 2 = 16, CDLL = -18071.7, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0051
Stage 2 = 17, CDLL = -18045.7, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0038
Stage 2 = 18, CDLL = -18009.6, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0045
Stage 2 = 19, CDLL = -18036.1, AR(1.16; 0.00) = [0.41; 0.07], Max-Change = 0.0031
Stage 2 = 20, CDLL = -18083.0, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0054
Stage 2 = 21, CDLL = -18060.5, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0013
Stage 2 = 22, CDLL = -18042.1, AR(1.16; 0.00) = [0.41; 0.03], Max-Change = 0.0015
Stage 2 = 23, CDLL = -18029.7, AR(1.16; 0.00) = [0.40; 0.07], Max-Change = 0.0024
Stage 2 = 24, CDLL = -18035.9, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0061
Stage 2 = 25, CDLL = -18031.5, AR(1.16; 0.00) = [0.41; 0.10], Max-Change = 0.0079
Stage 2 = 26, CDLL = -18033.7, AR(1.16; 0.00) = [0.41; 0.07], Max-Change = 0.0042
Stage 2 = 27, CDLL = -18030.0, AR(1.16; 0.00) = [0.37; 0.13], Max-Change = 0.0036
Stage 2 = 28, CDLL = -18031.7, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0095
Stage 2 = 29, CDLL = -18037.9, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0060
Stage 2 = 30, CDLL = -18041.0, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0069
Stage 2 = 31, CDLL = -18026.1, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0023
Stage 2 = 32, CDLL = -18039.2, AR(1.16; 0.00) = [0.41; 0.00], Max-Change = 0.0047
Stage 2 = 33, CDLL = -18037.1, AR(1.16; 0.00) = [0.42; 0.03], Max-Change = 0.0053
Stage 2 = 34, CDLL = -18024.4, AR(1.16; 0.00) = [0.40; 0.00], Max-Change = 0.0028
Stage 2 = 35, CDLL = -18036.3, AR(1.16; 0.00) = [0.42; 0.07], Max-Change = 0.0078
Stage 2 = 36, CDLL = -18012.7, AR(1.16; 0.00) = [0.41; 0.00], Max-Change = 0.0035
Stage 2 = 37, CDLL = -17989.1, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0025
Stage 2 = 38, CDLL = -18037.3, AR(1.16; 0.00) = [0.41; 0.03], Max-Change = 0.0045
Stage 2 = 39, CDLL = -18055.8, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0069
Stage 2 = 40, CDLL = -18043.3, AR(1.16; 0.00) = [0.42; 0.00], Max-Change = 0.0064
Stage 2 = 41, CDLL = -18004.3, AR(1.16; 0.00) = [0.42; 0.03], Max-Change = 0.0024
Stage 2 = 42, CDLL = -18006.0, AR(1.16; 0.00) = [0.41; 0.13], Max-Change = 0.0026
Stage 2 = 43, CDLL = -18017.3, AR(1.16; 0.00) = [0.41; 0.13], Max-Change = 0.0025
Stage 2 = 44, CDLL = -18008.1, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0053
Stage 2 = 45, CDLL = -18028.2, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0161
Stage 2 = 46, CDLL = -18042.6, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0060
Stage 2 = 47, CDLL = -18076.3, AR(1.16; 0.00) = [0.41; 0.03], Max-Change = 0.0077
Stage 2 = 48, CDLL = -17980.5, AR(1.16; 0.00) = [0.39; 0.13], Max-Change = 0.0065
Stage 2 = 49, CDLL = -18016.4, AR(1.16; 0.00) = [0.41; 0.00], Max-Change = 0.0030
Stage 2 = 50, CDLL = -18040.6, AR(1.16; 0.00) = [0.40; 0.10], Max-Change = 0.0060
Stage 2 = 51, CDLL = -18020.3, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0033
Stage 2 = 52, CDLL = -17975.7, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0037
Stage 2 = 53, CDLL = -18004.8, AR(1.16; 0.00) = [0.38; 0.03], Max-Change = 0.0104
Stage 2 = 54, CDLL = -18048.9, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0044
Stage 2 = 55, CDLL = -18020.2, AR(1.16; 0.00) = [0.40; 0.10], Max-Change = 0.0048
Stage 2 = 56, CDLL = -18034.2, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0065
Stage 2 = 57, CDLL = -18050.5, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0053
Stage 2 = 58, CDLL = -18021.4, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0053
Stage 2 = 59, CDLL = -18037.5, AR(1.16; 0.00) = [0.41; 0.10], Max-Change = 0.0041
Stage 2 = 60, CDLL = -18012.3, AR(1.16; 0.00) = [0.41; 0.13], Max-Change = 0.0048
Stage 2 = 61, CDLL = -18023.1, AR(1.16; 0.00) = [0.42; 0.07], Max-Change = 0.0051
Stage 2 = 62, CDLL = -18008.7, AR(1.16; 0.00) = [0.37; 0.03], Max-Change = 0.0055
Stage 2 = 63, CDLL = -18012.5, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0129
Stage 2 = 64, CDLL = -18051.2, AR(1.16; 0.00) = [0.38; 0.03], Max-Change = 0.0020
Stage 2 = 65, CDLL = -18073.7, AR(1.16; 0.00) = [0.41; 0.03], Max-Change = 0.0098
Stage 2 = 66, CDLL = -17993.5, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0048
Stage 2 = 67, CDLL = -18040.1, AR(1.16; 0.00) = [0.40; 0.07], Max-Change = 0.0044
Stage 2 = 68, CDLL = -18012.7, AR(1.16; 0.00) = [0.38; 0.10], Max-Change = 0.0054
Stage 2 = 69, CDLL = -18066.8, AR(1.16; 0.00) = [0.39; 0.00], Max-Change = 0.0028
Stage 2 = 70, CDLL = -18061.5, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0035
Stage 2 = 71, CDLL = -18015.0, AR(1.16; 0.00) = [0.37; 0.07], Max-Change = 0.0027
Stage 2 = 72, CDLL = -18022.0, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0043
Stage 2 = 73, CDLL = -18040.2, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0073
Stage 2 = 74, CDLL = -18047.5, AR(1.16; 0.00) = [0.39; 0.00], Max-Change = 0.0011
Stage 2 = 75, CDLL = -18035.2, AR(1.16; 0.00) = [0.42; 0.07], Max-Change = 0.0010
Stage 2 = 76, CDLL = -18059.9, AR(1.16; 0.00) = [0.41; 0.10], Max-Change = 0.0075
Stage 2 = 77, CDLL = -18011.0, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0092
Stage 2 = 78, CDLL = -18027.8, AR(1.16; 0.00) = [0.38; 0.13], Max-Change = 0.0035
Stage 2 = 79, CDLL = -18067.4, AR(1.16; 0.00) = [0.40; 0.07], Max-Change = 0.0054
Stage 2 = 80, CDLL = -18041.5, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0070
Stage 2 = 81, CDLL = -18055.5, AR(1.16; 0.00) = [0.40; 0.07], Max-Change = 0.0022
Stage 2 = 82, CDLL = -18049.1, AR(1.16; 0.00) = [0.38; 0.10], Max-Change = 0.0053
Stage 2 = 83, CDLL = -18014.9, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0053
Stage 2 = 84, CDLL = -18009.9, AR(1.16; 0.00) = [0.38; 0.10], Max-Change = 0.0142
Stage 2 = 85, CDLL = -18003.0, AR(1.16; 0.00) = [0.39; 0.03], Max-Change = 0.0049
Stage 2 = 86, CDLL = -17986.6, AR(1.16; 0.00) = [0.39; 0.07], Max-Change = 0.0045
Stage 2 = 87, CDLL = -18016.3, AR(1.16; 0.00) = [0.40; 0.10], Max-Change = 0.0013
Stage 2 = 88, CDLL = -18049.4, AR(1.16; 0.00) = [0.42; 0.03], Max-Change = 0.0004
Stage 2 = 89, CDLL = -18038.3, AR(1.16; 0.00) = [0.41; 0.07], Max-Change = 0.0080
Stage 2 = 90, CDLL = -18072.2, AR(1.16; 0.00) = [0.41; 0.03], Max-Change = 0.0101
Stage 2 = 91, CDLL = -18017.5, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0035
Stage 2 = 92, CDLL = -18059.5, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0105
Stage 2 = 93, CDLL = -18023.8, AR(1.16; 0.00) = [0.41; 0.03], Max-Change = 0.0044
Stage 2 = 94, CDLL = -18037.1, AR(1.16; 0.00) = [0.39; 0.10], Max-Change = 0.0034
Stage 2 = 95, CDLL = -18013.3, AR(1.16; 0.00) = [0.39; 0.17], Max-Change = 0.0023
Stage 2 = 96, CDLL = -18004.6, AR(1.16; 0.00) = [0.40; 0.13], Max-Change = 0.0027
Stage 2 = 97, CDLL = -18016.2, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0034
Stage 2 = 98, CDLL = -18025.8, AR(1.16; 0.00) = [0.37; 0.07], Max-Change = 0.0012
Stage 2 = 99, CDLL = -18027.2, AR(1.16; 0.00) = [0.40; 0.03], Max-Change = 0.0042
Stage 2 = 100, CDLL = -18027.6, AR(1.16; 0.00) = [0.38; 0.07], Max-Change = 0.0081
Stage 3 = 1, CDLL = -18061.1, AR(1.16; 0.00) = [0.40; 0.00], gam = 0.0000, Max-Change = 0.0000
Stage 3 = 2, CDLL = -18023.5, AR(1.16; 0.00) = [0.40; 0.07], gam = 0.1778, Max-Change = 0.0039
Stage 3 = 3, CDLL = -17999.7, AR(1.16; 0.00) = [0.41; 0.03], gam = 0.1057, Max-Change = 0.0051
Stage 3 = 4, CDLL = -18059.4, AR(1.16; 0.00) = [0.41; 0.07], gam = 0.0780, Max-Change = 0.0035
Stage 3 = 5, CDLL = -18023.9, AR(1.16; 0.00) = [0.38; 0.03], gam = 0.0629, Max-Change = 0.0013
Stage 3 = 6, CDLL = -18039.3, AR(1.16; 0.00) = [0.41; 0.00], gam = 0.0532, Max-Change = 0.0013
Stage 3 = 7, CDLL = -18041.4, AR(1.16; 0.00) = [0.41; 0.03], gam = 0.0464, Max-Change = 0.0024
Stage 3 = 8, CDLL = -18036.7, AR(1.16; 0.00) = [0.40; 0.00], gam = 0.0413, Max-Change = 0.0011
Stage 3 = 9, CDLL = -17997.2, AR(1.16; 0.00) = [0.38; 0.10], gam = 0.0374, Max-Change = 0.0002
Stage 3 = 10, CDLL = -18071.8, AR(1.16; 0.00) = [0.40; 0.00], gam = 0.0342, Max-Change = 0.0006
Stage 3 = 11, CDLL = -18042.5, AR(1.16; 0.00) = [0.41; 0.03], gam = 0.0316, Max-Change = 0.0003

#> 
#> Calculating log-likelihood...
coef(LLTM.e)
#> $Item_1
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_2
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_3
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_4
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_5
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_6
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_7
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_8
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_9
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_10
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_11
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_12
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_13
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_14
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_15
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_16
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_17
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_18
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_19
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_20
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_21
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_22
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_23
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_24
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_25
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_26
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_27
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_28
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_29
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $Item_30
#>     difficultyeasy difficultyhard difficultymedium a1 d g u
#> par          1.167         -0.928            0.047  1 0 0 1
#> 
#> $GroupPars
#>     MEAN_1 COV_11
#> par      0  0.965
#> 
#> $items
#>     COV_items_items
#> par           0.009
#> 


###################
# General MLTM example (Embretson, 1984)

set.seed(42)

as <- matrix(rep(1,60), ncol=2)
as[11:18,1] <- as[1:9,2] <- 0
d1 <- rep(c(3,1),each = 6)  # first easy, then medium, last difficult for first trait
d2 <- rep(c(0,1,2),times = 4)    # difficult to easy
d <- rnorm(18)
ds <- rbind(cbind(d1=NA, d2=d), cbind(d1, d2))
(pars <- data.frame(a=as, d=ds))
#>    a.1 a.2 d.d1        d.d2
#> 1    1   0   NA  1.37095845
#> 2    1   0   NA -0.56469817
#> 3    1   0   NA  0.36312841
#> 4    1   0   NA  0.63286260
#> 5    1   0   NA  0.40426832
#> 6    1   0   NA -0.10612452
#> 7    1   0   NA  1.51152200
#> 8    1   0   NA -0.09465904
#> 9    1   0   NA  2.01842371
#> 10   1   1   NA -0.06271410
#> 11   0   1   NA  1.30486965
#> 12   0   1   NA  2.28664539
#> 13   0   1   NA -1.38886070
#> 14   0   1   NA -0.27878877
#> 15   0   1   NA -0.13332134
#> 16   0   1   NA  0.63595040
#> 17   0   1   NA -0.28425292
#> 18   0   1   NA -2.65645542
#> 19   1   1    3  0.00000000
#> 20   1   1    3  1.00000000
#> 21   1   1    3  2.00000000
#> 22   1   1    3  0.00000000
#> 23   1   1    3  1.00000000
#> 24   1   1    3  2.00000000
#> 25   1   1    1  0.00000000
#> 26   1   1    1  1.00000000
#> 27   1   1    1  2.00000000
#> 28   1   1    1  0.00000000
#> 29   1   1    1  1.00000000
#> 30   1   1    1  2.00000000
dat <- simdata(as, ds, 2500,
  itemtype = c(rep('dich', 18), rep('partcomp', 12)))
itemstats(dat)
#> $overall
#>     N mean_total.score sd_total.score ave.r  sd.r alpha SEM.alpha
#>  2500           16.494           4.83 0.088 0.059 0.747     2.428
#> 
#> $itemstats
#>            N  mean    sd total.r total.r_if_rm alpha_if_rm
#> Item_1  2500 0.752 0.432   0.265         0.180       0.745
#> Item_2  2500 0.384 0.486   0.328         0.234       0.742
#> Item_3  2500 0.563 0.496   0.319         0.222       0.743
#> Item_4  2500 0.635 0.481   0.318         0.224       0.743
#> Item_5  2500 0.582 0.493   0.320         0.224       0.743
#> Item_6  2500 0.478 0.500   0.329         0.233       0.742
#> Item_7  2500 0.767 0.423   0.274         0.191       0.744
#> Item_8  2500 0.469 0.499   0.315         0.218       0.743
#> Item_9  2500 0.849 0.358   0.233         0.161       0.745
#> Item_10 2500 0.471 0.499   0.557         0.480       0.727
#> Item_11 2500 0.736 0.441   0.352         0.268       0.740
#> Item_12 2500 0.882 0.323   0.246         0.182       0.745
#> Item_13 2500 0.232 0.422   0.302         0.220       0.743
#> Item_14 2500 0.460 0.499   0.319         0.222       0.743
#> Item_15 2500 0.480 0.500   0.387         0.294       0.739
#> Item_16 2500 0.627 0.484   0.352         0.260       0.741
#> Item_17 2500 0.441 0.497   0.318         0.222       0.743
#> Item_18 2500 0.097 0.296   0.209         0.149       0.746
#> Item_19 2500 0.466 0.499   0.381         0.287       0.739
#> Item_20 2500 0.643 0.479   0.360         0.269       0.740
#> Item_21 2500 0.788 0.409   0.335         0.257       0.741
#> Item_22 2500 0.456 0.498   0.406         0.315       0.737
#> Item_23 2500 0.646 0.478   0.403         0.315       0.737
#> Item_24 2500 0.769 0.422   0.364         0.284       0.740
#> Item_25 2500 0.349 0.477   0.408         0.321       0.737
#> Item_26 2500 0.492 0.500   0.414         0.323       0.737
#> Item_27 2500 0.586 0.493   0.381         0.289       0.739
#> Item_28 2500 0.330 0.470   0.388         0.300       0.738
#> Item_29 2500 0.477 0.500   0.361         0.266       0.740
#> Item_30 2500 0.587 0.492   0.371         0.278       0.740
#> 
#> $proportions
#>             0     1
#> Item_1  0.248 0.752
#> Item_2  0.616 0.384
#> Item_3  0.437 0.563
#> Item_4  0.365 0.635
#> Item_5  0.418 0.582
#> Item_6  0.522 0.478
#> Item_7  0.233 0.767
#> Item_8  0.531 0.469
#> Item_9  0.151 0.849
#> Item_10 0.529 0.471
#> Item_11 0.264 0.736
#> Item_12 0.118 0.882
#> Item_13 0.768 0.232
#> Item_14 0.540 0.460
#> Item_15 0.520 0.480
#> Item_16 0.373 0.627
#> Item_17 0.559 0.441
#> Item_18 0.903 0.097
#> Item_19 0.534 0.466
#> Item_20 0.357 0.643
#> Item_21 0.212 0.788
#> Item_22 0.544 0.456
#> Item_23 0.354 0.646
#> Item_24 0.231 0.769
#> Item_25 0.651 0.349
#> Item_26 0.508 0.492
#> Item_27 0.414 0.586
#> Item_28 0.670 0.330
#> Item_29 0.523 0.477
#> Item_30 0.413 0.587
#> 

# unconditional model
syntax <- "theta1 = 1-9, 19-30
           theta2 = 10-30
           COV = theta1*theta2"
itemtype <- c(rep('Rasch', 18), rep('PC1PL', 12))
mod <- mirt(dat, syntax, itemtype=itemtype)
#> 
Iteration: 1, Log-Lik: -45647.618, Max-Change: 1.61713
Iteration: 2, Log-Lik: -43934.319, Max-Change: 0.56278
Iteration: 3, Log-Lik: -43892.332, Max-Change: 0.38750
Iteration: 4, Log-Lik: -43877.383, Max-Change: 0.26104
Iteration: 5, Log-Lik: -43870.300, Max-Change: 0.27835
Iteration: 6, Log-Lik: -43866.174, Max-Change: 0.07772
Iteration: 7, Log-Lik: -43864.938, Max-Change: 0.13203
Iteration: 8, Log-Lik: -43863.307, Max-Change: 0.08207
Iteration: 9, Log-Lik: -43862.326, Max-Change: 0.06080
Iteration: 10, Log-Lik: -43861.278, Max-Change: 0.08802
Iteration: 11, Log-Lik: -43860.960, Max-Change: 0.17156
Iteration: 12, Log-Lik: -43860.658, Max-Change: 0.09321
Iteration: 13, Log-Lik: -43860.473, Max-Change: 0.08516
Iteration: 14, Log-Lik: -43860.372, Max-Change: 0.10367
Iteration: 15, Log-Lik: -43860.296, Max-Change: 0.03893
Iteration: 16, Log-Lik: -43860.263, Max-Change: 0.04451
Iteration: 17, Log-Lik: -43860.232, Max-Change: 0.03615
Iteration: 18, Log-Lik: -43860.211, Max-Change: 0.03300
Iteration: 19, Log-Lik: -43860.188, Max-Change: 0.00524
Iteration: 20, Log-Lik: -43860.173, Max-Change: 0.00129
Iteration: 21, Log-Lik: -43860.170, Max-Change: 0.00080
Iteration: 22, Log-Lik: -43860.169, Max-Change: 0.00052
Iteration: 23, Log-Lik: -43860.168, Max-Change: 0.00069
Iteration: 24, Log-Lik: -43860.168, Max-Change: 0.00064
Iteration: 25, Log-Lik: -43860.168, Max-Change: 0.00132
Iteration: 26, Log-Lik: -43860.167, Max-Change: 0.00048
Iteration: 27, Log-Lik: -43860.167, Max-Change: 0.00010
Iteration: 28, Log-Lik: -43860.167, Max-Change: 0.00012
Iteration: 29, Log-Lik: -43860.167, Max-Change: 0.00014
Iteration: 30, Log-Lik: -43860.167, Max-Change: 0.00014
Iteration: 31, Log-Lik: -43860.167, Max-Change: 0.00007
coef(mod, simplify=TRUE)
#> $items
#>         a1 a2      d g u    d1     d2
#> Item_1   1  0  1.313 0 1    NA     NA
#> Item_2   1  0 -0.563 0 1    NA     NA
#> Item_3   1  0  0.303 0 1    NA     NA
#> Item_4   1  0  0.660 0 1    NA     NA
#> Item_5   1  0  0.393 0 1    NA     NA
#> Item_6   1  0 -0.105 0 1    NA     NA
#> Item_7   1  0  1.404 0 1    NA     NA
#> Item_8   1  0 -0.147 0 1    NA     NA
#> Item_9   1  0  2.013 0 1    NA     NA
#> Item_10  0  1 -0.141 0 1    NA     NA
#> Item_11  0  1  1.227 0 1    NA     NA
#> Item_12  0  1  2.350 0 1    NA     NA
#> Item_13  0  1 -1.429 0 1    NA     NA
#> Item_14  0  1 -0.193 0 1    NA     NA
#> Item_15  0  1 -0.098 0 1    NA     NA
#> Item_16  0  1  0.623 0 1    NA     NA
#> Item_17  0  1 -0.286 0 1    NA     NA
#> Item_18  0  1 -2.592 0 1    NA     NA
#> Item_19  1  1     NA 0 1 2.870  0.013
#> Item_20  1  1     NA 0 1 3.716  0.832
#> Item_21  1  1     NA 0 1 3.238  1.900
#> Item_22  1  1     NA 0 1 4.407 -0.174
#> Item_23  1  1     NA 0 1 3.538  0.866
#> Item_24  1  1     NA 0 1 2.851  1.890
#> Item_25  1  1     NA 0 1 1.197 -0.137
#> Item_26  1  1     NA 0 1 1.038  0.975
#> Item_27  1  1     NA 0 1 1.063  1.818
#> Item_28  1  1     NA 0 1 0.970 -0.138
#> Item_29  1  1     NA 0 1 0.902  1.010
#> Item_30  1  1     NA 0 1 1.015  1.914
#> 
#> $means
#> theta1 theta2 
#>      0      0 
#> 
#> $cov
#>        theta1 theta2
#> theta1  0.917  0.081
#> theta2  0.081  0.984
#> 
data.frame(est=coef(mod, simplify=TRUE)$items, pop=data.frame(a=as, d=ds))
#>         est.a1 est.a2       est.d est.g est.u    est.d1      est.d2 pop.a.1
#> Item_1       1      0  1.31307855     0     1        NA          NA       1
#> Item_2       1      0 -0.56331423     0     1        NA          NA       1
#> Item_3       1      0  0.30311181     0     1        NA          NA       1
#> Item_4       1      0  0.66017201     0     1        NA          NA       1
#> Item_5       1      0  0.39267181     0     1        NA          NA       1
#> Item_6       1      0 -0.10529560     0     1        NA          NA       1
#> Item_7       1      0  1.40429460     0     1        NA          NA       1
#> Item_8       1      0 -0.14742905     0     1        NA          NA       1
#> Item_9       1      0  2.01310778     0     1        NA          NA       1
#> Item_10      0      1 -0.14050425     0     1        NA          NA       1
#> Item_11      0      1  1.22727159     0     1        NA          NA       0
#> Item_12      0      1  2.35002102     0     1        NA          NA       0
#> Item_13      0      1 -1.42944927     0     1        NA          NA       0
#> Item_14      0      1 -0.19283592     0     1        NA          NA       0
#> Item_15      0      1 -0.09794928     0     1        NA          NA       0
#> Item_16      0      1  0.62283206     0     1        NA          NA       0
#> Item_17      0      1 -0.28628773     0     1        NA          NA       0
#> Item_18      0      1 -2.59213954     0     1        NA          NA       0
#> Item_19      1      1          NA     0     1 2.8695796  0.01318001       1
#> Item_20      1      1          NA     0     1 3.7157820  0.83154537       1
#> Item_21      1      1          NA     0     1 3.2383233  1.90018431       1
#> Item_22      1      1          NA     0     1 4.4073154 -0.17448556       1
#> Item_23      1      1          NA     0     1 3.5382631  0.86641897       1
#> Item_24      1      1          NA     0     1 2.8505117  1.88958564       1
#> Item_25      1      1          NA     0     1 1.1972230 -0.13675332       1
#> Item_26      1      1          NA     0     1 1.0378260  0.97497271       1
#> Item_27      1      1          NA     0     1 1.0634033  1.81828381       1
#> Item_28      1      1          NA     0     1 0.9704764 -0.13769090       1
#> Item_29      1      1          NA     0     1 0.9017976  1.00959896       1
#> Item_30      1      1          NA     0     1 1.0149521  1.91386682       1
#>         pop.a.2 pop.d.d1    pop.d.d2
#> Item_1        0       NA  1.37095845
#> Item_2        0       NA -0.56469817
#> Item_3        0       NA  0.36312841
#> Item_4        0       NA  0.63286260
#> Item_5        0       NA  0.40426832
#> Item_6        0       NA -0.10612452
#> Item_7        0       NA  1.51152200
#> Item_8        0       NA -0.09465904
#> Item_9        0       NA  2.01842371
#> Item_10       1       NA -0.06271410
#> Item_11       1       NA  1.30486965
#> Item_12       1       NA  2.28664539
#> Item_13       1       NA -1.38886070
#> Item_14       1       NA -0.27878877
#> Item_15       1       NA -0.13332134
#> Item_16       1       NA  0.63595040
#> Item_17       1       NA -0.28425292
#> Item_18       1       NA -2.65645542
#> Item_19       1        3  0.00000000
#> Item_20       1        3  1.00000000
#> Item_21       1        3  2.00000000
#> Item_22       1        3  0.00000000
#> Item_23       1        3  1.00000000
#> Item_24       1        3  2.00000000
#> Item_25       1        1  0.00000000
#> Item_26       1        1  1.00000000
#> Item_27       1        1  2.00000000
#> Item_28       1        1  0.00000000
#> Item_29       1        1  1.00000000
#> Item_30       1        1  2.00000000
itemplot(mod, 1)

itemplot(mod, 30)


# MLTM design only for PC1PL items
itemdesign <- data.frame(t1_difficulty= factor(d1, labels=c('medium', 'easy')),
                        t2_difficulty=factor(d2, labels=c('hard', 'medium', 'easy')))
rownames(itemdesign) <- colnames(dat)[19:30]
itemdesign
#>         t1_difficulty t2_difficulty
#> Item_19          easy          hard
#> Item_20          easy        medium
#> Item_21          easy          easy
#> Item_22          easy          hard
#> Item_23          easy        medium
#> Item_24          easy          easy
#> Item_25        medium          hard
#> Item_26        medium        medium
#> Item_27        medium          easy
#> Item_28        medium          hard
#> Item_29        medium        medium
#> Item_30        medium          easy

# fit MLTM design, leaving first 18 items as 'Rasch' type
mltm <- mirt(dat, syntax, itemtype=itemtype, itemdesign=itemdesign,
             item.formula = list(theta1 ~ 0 + t1_difficulty,
                                 theta2 ~ 0 + t2_difficulty), SE=TRUE)
#> 
Iteration: 1, Log-Lik: -49303.248, Max-Change: 1.93957
Iteration: 2, Log-Lik: -44367.201, Max-Change: 0.45231
Iteration: 3, Log-Lik: -44219.190, Max-Change: 0.27074
Iteration: 4, Log-Lik: -44116.596, Max-Change: 0.17330
Iteration: 5, Log-Lik: -44038.884, Max-Change: 0.10667
Iteration: 6, Log-Lik: -43981.731, Max-Change: 0.07106
Iteration: 7, Log-Lik: -43941.387, Max-Change: 0.04798
Iteration: 8, Log-Lik: -43914.195, Max-Change: 0.03703
Iteration: 9, Log-Lik: -43896.604, Max-Change: 0.03031
Iteration: 10, Log-Lik: -43885.583, Max-Change: 0.02433
Iteration: 11, Log-Lik: -43878.831, Max-Change: 0.01923
Iteration: 12, Log-Lik: -43874.753, Max-Change: 0.01512
Iteration: 13, Log-Lik: -43869.120, Max-Change: 0.00606
Iteration: 14, Log-Lik: -43868.905, Max-Change: 0.00385
Iteration: 15, Log-Lik: -43868.798, Max-Change: 0.00302
Iteration: 16, Log-Lik: -43868.694, Max-Change: 0.00277
Iteration: 17, Log-Lik: -43868.674, Max-Change: 0.00181
Iteration: 18, Log-Lik: -43868.665, Max-Change: 0.00133
Iteration: 19, Log-Lik: -43868.658, Max-Change: 0.00114
Iteration: 20, Log-Lik: -43868.656, Max-Change: 0.00071
Iteration: 21, Log-Lik: -43868.655, Max-Change: 0.00051
Iteration: 22, Log-Lik: -43868.654, Max-Change: 0.00043
Iteration: 23, Log-Lik: -43868.654, Max-Change: 0.00027
Iteration: 24, Log-Lik: -43868.654, Max-Change: 0.00019
Iteration: 25, Log-Lik: -43868.653, Max-Change: 0.00016
Iteration: 26, Log-Lik: -43868.653, Max-Change: 0.00010
#> 
#> Calculating information matrix...
coef(mltm, simplify=TRUE)
#> $items
#>         theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> Item_1                      0.00                      0.000
#> Item_2                      0.00                      0.000
#> Item_3                      0.00                      0.000
#> Item_4                      0.00                      0.000
#> Item_5                      0.00                      0.000
#> Item_6                      0.00                      0.000
#> Item_7                      0.00                      0.000
#> Item_8                      0.00                      0.000
#> Item_9                      0.00                      0.000
#> Item_10                     0.00                      0.000
#> Item_11                     0.00                      0.000
#> Item_12                     0.00                      0.000
#> Item_13                     0.00                      0.000
#> Item_14                     0.00                      0.000
#> Item_15                     0.00                      0.000
#> Item_16                     0.00                      0.000
#> Item_17                     0.00                      0.000
#> Item_18                     0.00                      0.000
#> Item_19                     3.19                      0.000
#> Item_20                     3.19                      0.000
#> Item_21                     3.19                      0.000
#> Item_22                     3.19                      0.000
#> Item_23                     3.19                      0.000
#> Item_24                     3.19                      0.000
#> Item_25                     0.00                      1.031
#> Item_26                     0.00                      1.031
#> Item_27                     0.00                      1.031
#> Item_28                     0.00                      1.031
#> Item_29                     0.00                      1.031
#> Item_30                     0.00                      1.031
#>         theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> Item_1                     0.000                    0.000
#> Item_2                     0.000                    0.000
#> Item_3                     0.000                    0.000
#> Item_4                     0.000                    0.000
#> Item_5                     0.000                    0.000
#> Item_6                     0.000                    0.000
#> Item_7                     0.000                    0.000
#> Item_8                     0.000                    0.000
#> Item_9                     0.000                    0.000
#> Item_10                    0.000                    0.000
#> Item_11                    0.000                    0.000
#> Item_12                    0.000                    0.000
#> Item_13                    0.000                    0.000
#> Item_14                    0.000                    0.000
#> Item_15                    0.000                    0.000
#> Item_16                    0.000                    0.000
#> Item_17                    0.000                    0.000
#> Item_18                    0.000                    0.000
#> Item_19                    0.000                   -0.078
#> Item_20                    0.000                    0.000
#> Item_21                    1.857                    0.000
#> Item_22                    0.000                   -0.078
#> Item_23                    0.000                    0.000
#> Item_24                    1.857                    0.000
#> Item_25                    0.000                   -0.078
#> Item_26                    0.000                    0.000
#> Item_27                    1.857                    0.000
#> Item_28                    0.000                   -0.078
#> Item_29                    0.000                    0.000
#> Item_30                    1.857                    0.000
#>         theta2.t2_difficultymedium a1 a2      d g u d1 d2
#> Item_1                       0.000  1  0  1.314 0 1 NA NA
#> Item_2                       0.000  1  0 -0.563 0 1 NA NA
#> Item_3                       0.000  1  0  0.303 0 1 NA NA
#> Item_4                       0.000  1  0  0.661 0 1 NA NA
#> Item_5                       0.000  1  0  0.393 0 1 NA NA
#> Item_6                       0.000  1  0 -0.105 0 1 NA NA
#> Item_7                       0.000  1  0  1.405 0 1 NA NA
#> Item_8                       0.000  1  0 -0.147 0 1 NA NA
#> Item_9                       0.000  1  0  2.014 0 1 NA NA
#> Item_10                      0.000  0  1 -0.140 0 1 NA NA
#> Item_11                      0.000  0  1  1.228 0 1 NA NA
#> Item_12                      0.000  0  1  2.351 0 1 NA NA
#> Item_13                      0.000  0  1 -1.430 0 1 NA NA
#> Item_14                      0.000  0  1 -0.193 0 1 NA NA
#> Item_15                      0.000  0  1 -0.098 0 1 NA NA
#> Item_16                      0.000  0  1  0.623 0 1 NA NA
#> Item_17                      0.000  0  1 -0.286 0 1 NA NA
#> Item_18                      0.000  0  1 -2.594 0 1 NA NA
#> Item_19                      0.000  1  1     NA 0 1  0  0
#> Item_20                      0.924  1  1     NA 0 1  0  0
#> Item_21                      0.000  1  1     NA 0 1  0  0
#> Item_22                      0.000  1  1     NA 0 1  0  0
#> Item_23                      0.924  1  1     NA 0 1  0  0
#> Item_24                      0.000  1  1     NA 0 1  0  0
#> Item_25                      0.000  1  1     NA 0 1  0  0
#> Item_26                      0.924  1  1     NA 0 1  0  0
#> Item_27                      0.000  1  1     NA 0 1  0  0
#> Item_28                      0.000  1  1     NA 0 1  0  0
#> Item_29                      0.924  1  1     NA 0 1  0  0
#> Item_30                      0.000  1  1     NA 0 1  0  0
#> 
#> $means
#> theta1 theta2 
#>      0      0 
#> 
#> $cov
#>        theta1 theta2
#> theta1  0.919  0.074
#> theta2  0.074  0.988
#> 
coef(mltm, printSE=TRUE)
#> $Item_1
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  1  0 1.314     -999      999
#> SE                          NA NA NA 0.054       NA       NA
#> 
#> $Item_2
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  1  0 -0.563     -999      999
#> SE                          NA NA NA  0.049       NA       NA
#> 
#> $Item_3
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  1  0 0.303     -999      999
#> SE                          NA NA NA 0.048       NA       NA
#> 
#> $Item_4
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  1  0 0.661     -999      999
#> SE                          NA NA NA 0.049       NA       NA
#> 
#> $Item_5
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  1  0 0.393     -999      999
#> SE                          NA NA NA 0.048       NA       NA
#> 
#> $Item_6
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  1  0 -0.105     -999      999
#> SE                          NA NA NA  0.048       NA       NA
#> 
#> $Item_7
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  1  0 1.405     -999      999
#> SE                          NA NA NA 0.055       NA       NA
#> 
#> $Item_8
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  1  0 -0.147     -999      999
#> SE                          NA NA NA  0.048       NA       NA
#> 
#> $Item_9
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  1  0 2.014     -999      999
#> SE                          NA NA NA 0.063       NA       NA
#> 
#> $Item_10
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  0  1 -0.140     -999      999
#> SE                          NA NA NA  0.048       NA       NA
#> 
#> $Item_11
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  0  1 1.228     -999      999
#> SE                          NA NA NA 0.053       NA       NA
#> 
#> $Item_12
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  0  1 2.351     -999      999
#> SE                          NA NA NA 0.069       NA       NA
#> 
#> $Item_13
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  0  1 -1.430     -999      999
#> SE                          NA NA NA  0.055       NA       NA
#> 
#> $Item_14
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  0  1 -0.193     -999      999
#> SE                          NA NA NA  0.048       NA       NA
#> 
#> $Item_15
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  0  1 -0.098     -999      999
#> SE                          NA NA NA  0.048       NA       NA
#> 
#> $Item_16
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2     d logit(g) logit(u)
#> par                          0  0  1 0.623     -999      999
#> SE                          NA NA NA 0.050       NA       NA
#> 
#> $Item_17
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  0  1 -0.286     -999      999
#> SE                          NA NA NA  0.049       NA       NA
#> 
#> $Item_18
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                        0                          0
#> SE                        NA                         NA
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                        0                        0
#> SE                        NA                       NA
#>     theta2.t2_difficultymedium a1 a2      d logit(g) logit(u)
#> par                          0  0  1 -2.594     -999      999
#> SE                          NA NA NA  0.074       NA       NA
#> 
#> $Item_19
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_20
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_21
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_22
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_23
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_24
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_25
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_26
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_27
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_28
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_29
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $Item_30
#>     theta1.t1_difficultyeasy theta1.t1_difficultymedium
#> par                    3.190                      1.031
#> SE                     0.145                      0.048
#>     theta2.t2_difficultyeasy theta2.t2_difficultyhard
#> par                    1.857                   -0.078
#> SE                     0.065                    0.037
#>     theta2.t2_difficultymedium a1 a2 d1 d2 logit(g) logit(u)
#> par                      0.924  1  1  0  0     -999      999
#> SE                       0.046 NA NA NA NA       NA       NA
#> 
#> $GroupPars
#>     MEAN_1 MEAN_2 COV_11 COV_21 COV_22
#> par      0      0  0.919  0.074  0.988
#> SE      NA     NA  0.047  0.029  0.045
#> 
anova(mltm, mod) # similar fit; hence more constrained version preferred
#>           AIC    SABIC       HQ      BIC    logLik     X2 df     p
#> mltm 87789.31 87858.12 87844.28 87940.73 -43868.65                
#> mod  87810.34 87929.44 87905.48 88072.42 -43860.17 16.972 19 0.592
M2(mltm) # goodness of fit
#>             M2  df            p      RMSEA    RMSEA_5   RMSEA_95      SRMSR
#> stats 724.3175 439 2.220446e-16 0.01612682 0.01400712 0.01819185 0.03054497
#>             TLI       CFI
#> stats 0.9760504 0.9758302
head(personfit(mltm))
#>      outfit   z.outfit     infit    z.infit         Zh
#> 1 0.4099102 -2.3020948 0.5059336 -2.9261983  2.2862274
#> 2 1.9123368  2.2520109 1.2180485  1.0684326 -1.5047067
#> 3 0.6286135 -0.8426793 0.7783421 -0.9888073  1.0049591
#> 4 0.7758581 -0.9554199 0.8563428 -0.8899493  0.9411033
#> 5 0.7022093 -0.8066740 0.8190887 -0.9254922  0.9442138
#> 6 0.4515079 -1.3679137 0.5692678 -1.6827425  1.4501646
residuals(mltm)
#> LD matrix (lower triangle) and standardized residual correlations (upper triangle)
#> 
#> Upper triangle summary:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>  -0.064  -0.024  -0.007   0.000   0.019   0.174 
#> 
#>         Item_1 Item_2 Item_3 Item_4 Item_5 Item_6 Item_7 Item_8 Item_9 Item_10
#> Item_1         -0.010 -0.027 -0.040 -0.031  0.029  0.000  0.003 -0.012   0.098
#> Item_2   0.229         0.008  0.013  0.041  0.049  0.015  0.003 -0.004   0.142
#> Item_3   1.788  0.156        -0.025 -0.007 -0.012  0.012 -0.014 -0.015   0.174
#> Item_4   3.964  0.437  1.577        -0.006  0.014 -0.014  0.008  0.018   0.134
#> Item_5   2.393  4.302  0.134  0.085         0.021  0.010 -0.015 -0.005   0.154
#> Item_6   2.036  5.900  0.344  0.505  1.137         0.005 -0.012 -0.003   0.127
#> Item_7   0.000  0.581  0.363  0.498  0.261  0.064         0.005  0.006   0.134
#> Item_8   0.029  0.019  0.490  0.155  0.560  0.341  0.075        -0.011   0.166
#> Item_9   0.389  0.035  0.534  0.827  0.072  0.029  0.097  0.280          0.093
#> Item_10 23.809 50.189 75.667 44.656 59.644 40.070 45.056 68.776 21.841        
#> Item_11  0.009  0.716  0.194  0.626  0.034  0.175  0.028  0.385  0.126   1.843
#> Item_12  0.753  0.236  3.918  0.095  1.109  0.582  0.818  1.181  5.494   0.011
#> Item_13  1.267  4.570  0.152  0.017  0.396  2.507  3.411  0.105  2.038   3.111
#> Item_14  7.361  0.214  0.172  0.934  1.965  4.761  4.570  0.319  3.784   1.714
#> Item_15  0.081  0.605  1.939  0.315  0.397  0.037  0.590  0.408  0.605   0.871
#> Item_16  1.749  0.256  2.779  0.965  1.745  0.134  2.493  0.001  3.477   0.281
#> Item_17  3.492  0.001  0.804  0.708  0.094  2.145  3.168  6.440  0.076   6.330
#> Item_18  1.296  0.205  0.068  0.911  0.003  0.029  0.301  0.309  0.158   0.447
#> Item_19  1.950  1.095  0.940  1.227  3.901  2.358  0.879  2.427  1.078   0.912
#> Item_20  0.821  1.556  0.836  0.291  0.071  0.862  1.370  0.069  6.617   0.072
#> Item_21  3.887  1.725  3.272  1.120  1.616  0.749  1.266  0.857  0.744   4.508
#> Item_22  2.670  0.247  0.016  4.552  0.345  0.736  0.650  0.010  0.030   1.904
#> Item_23  1.723  2.043  0.011  0.415  0.958  0.032  0.670  0.042  0.005   3.678
#> Item_24  6.039  2.418  3.100  6.560  2.357  5.733  2.198  2.200  4.635   7.501
#> Item_25  0.562  0.795  0.373  4.838  1.043  3.118  0.468  1.310  0.631  11.477
#> Item_26  4.256  1.169  0.889  0.700  1.184  0.633  3.494  1.488  3.177  20.873
#> Item_27  0.025  0.170  0.067  0.374  2.965  0.050  0.179  0.187  2.212  30.192
#> Item_28  2.586  7.102  3.183  2.042  2.136  2.200  2.730  3.809  2.120  13.578
#> Item_29  1.392  0.557  1.223  0.458  0.710  1.588  2.200  0.723  3.080  11.310
#> Item_30  0.831  1.556  1.214  0.302  0.109  0.449  0.573  0.340  0.424  20.892
#>         Item_11 Item_12 Item_13 Item_14 Item_15 Item_16 Item_17 Item_18 Item_19
#> Item_1    0.002  -0.017  -0.023  -0.054  -0.006  -0.026  -0.037  -0.023  -0.028
#> Item_2   -0.017  -0.010  -0.043  -0.009  -0.016  -0.010   0.000  -0.009   0.021
#> Item_3    0.009  -0.040   0.008  -0.008   0.028  -0.033  -0.018  -0.005  -0.019
#> Item_4   -0.016  -0.006  -0.003  -0.019  -0.011  -0.020  -0.017   0.019   0.022
#> Item_5   -0.004  -0.021  -0.013  -0.028  -0.013  -0.026   0.006  -0.001   0.040
#> Item_6    0.008  -0.015  -0.032  -0.044   0.004   0.007  -0.029  -0.003   0.031
#> Item_7    0.003   0.018  -0.037  -0.043  -0.015  -0.032  -0.036   0.011   0.019
#> Item_8    0.012  -0.022   0.006  -0.011   0.013   0.001  -0.051  -0.011   0.031
#> Item_9    0.007  -0.047  -0.029  -0.039  -0.016  -0.037   0.006   0.008   0.021
#> Item_10   0.027   0.002   0.035  -0.026   0.019   0.011  -0.050   0.013  -0.019
#> Item_11          -0.022   0.022  -0.006   0.009  -0.009  -0.027   0.006  -0.019
#> Item_12   1.214          -0.008  -0.027   0.011  -0.003  -0.014  -0.028  -0.020
#> Item_13   1.222   0.141          -0.013   0.035   0.017  -0.008  -0.027  -0.023
#> Item_14   0.096   1.802   0.404          -0.014   0.018  -0.013  -0.038  -0.042
#> Item_15   0.208   0.293   3.129   0.460           0.021  -0.011  -0.026   0.029
#> Item_16   0.208   0.018   0.759   0.815   1.060          -0.007  -0.003  -0.019
#> Item_17   1.773   0.461   0.180   0.392   0.290   0.140           0.005  -0.023
#> Item_18   0.100   1.944   1.860   3.653   1.691   0.020   0.064          -0.033
#> Item_19   0.913   1.015   1.284   4.468   2.033   0.859   1.304   2.695        
#> Item_20   1.446   0.039   0.045   2.772   4.411   0.072   0.042   3.259   0.949
#> Item_21   0.747   0.875   1.773   1.730   0.759   1.420   0.979   1.137   5.069
#> Item_22   0.717   0.201   0.020   0.388   5.866   0.030   0.071   1.091   1.009
#> Item_23   8.564   1.754   4.656   0.443   1.035   0.005   0.173   1.152   0.997
#> Item_24   3.006   2.617   2.265   2.175   2.826   2.329   2.261   4.236   6.280
#> Item_25   0.480   0.827   0.946   0.373   0.576   1.593   2.431   1.596   1.146
#> Item_26   1.356   0.825   1.879   2.200   0.954   1.012   5.500   4.288   1.332
#> Item_27   0.144   1.132   0.112   0.783   0.837   1.936   4.920   3.868   1.314
#> Item_28   2.097   4.981   2.166   2.329   2.583   3.338   2.228   2.049   3.566
#> Item_29   0.736   0.533   1.869   2.544   2.516   1.394  10.093   0.484   5.562
#> Item_30   0.221   0.086   5.871   2.007   0.460   0.201   5.445   2.127   0.862
#>         Item_20 Item_21 Item_22 Item_23 Item_24 Item_25 Item_26 Item_27 Item_28
#> Item_1   -0.018   0.039  -0.033  -0.026   0.049   0.015   0.041   0.003  -0.032
#> Item_2   -0.025  -0.026   0.010   0.029   0.031   0.018   0.022  -0.008   0.053
#> Item_3   -0.018  -0.036   0.003   0.002   0.035   0.012   0.019   0.005  -0.036
#> Item_4   -0.011   0.021   0.043   0.013   0.051  -0.044   0.017   0.012  -0.029
#> Item_5   -0.005  -0.025   0.012  -0.020   0.031  -0.020   0.022  -0.034   0.029
#> Item_6    0.019  -0.017  -0.017   0.004   0.048   0.035  -0.016  -0.004   0.030
#> Item_7   -0.023   0.023  -0.016  -0.016   0.030   0.014  -0.037  -0.008   0.033
#> Item_8    0.005  -0.019   0.002   0.004   0.030  -0.023  -0.024   0.009  -0.039
#> Item_9   -0.051  -0.017  -0.003  -0.001   0.043  -0.016  -0.036   0.030   0.029
#> Item_10  -0.005   0.042   0.028   0.038   0.055   0.068   0.091   0.110   0.074
#> Item_11   0.024  -0.017  -0.017   0.059   0.035  -0.014   0.023   0.008   0.029
#> Item_12  -0.004  -0.019   0.009   0.026  -0.032   0.018   0.018   0.021  -0.045
#> Item_13  -0.004  -0.027  -0.003   0.043  -0.030  -0.019  -0.027   0.007  -0.029
#> Item_14  -0.033  -0.026  -0.012  -0.013   0.029  -0.012  -0.030  -0.018  -0.031
#> Item_15   0.042  -0.017   0.048   0.020  -0.034   0.015  -0.020   0.018  -0.032
#> Item_16   0.005  -0.024   0.003   0.001   0.031   0.025   0.020  -0.028  -0.037
#> Item_17  -0.004   0.020   0.005  -0.008  -0.030  -0.031  -0.047  -0.044  -0.030
#> Item_18  -0.036   0.021   0.021  -0.021   0.041  -0.025  -0.041  -0.039  -0.029
#> Item_19   0.019  -0.045   0.020   0.020  -0.050   0.021  -0.023  -0.023  -0.038
#> Item_20           0.021   0.029   0.011  -0.045  -0.015  -0.019  -0.026  -0.036
#> Item_21   1.097          -0.018   0.022   0.041   0.031  -0.023   0.027   0.041
#> Item_22   2.114   0.778           0.023   0.046   0.035   0.016   0.005  -0.033
#> Item_23   0.304   1.176   1.318           0.036  -0.019   0.022  -0.003   0.037
#> Item_24   5.068   4.269   5.374   3.166           0.034   0.050  -0.030  -0.047
#> Item_25   0.569   2.467   3.060   0.871   2.822           0.019   0.020  -0.039
#> Item_26   0.944   1.268   0.663   1.248   6.316   0.862           0.032   0.036
#> Item_27   1.693   1.795   0.068   0.021   2.264   0.969   2.522          -0.033
#> Item_28   3.193   4.110   2.716   3.385   5.434   3.775   3.228   2.749        
#> Item_29   0.841   1.320   2.280   2.795   4.367   1.522   2.260   5.717   4.057
#> Item_30   6.126   0.904   0.633   3.421   3.815   0.664   1.371   0.140   3.142
#>         Item_29 Item_30
#> Item_1   -0.024  -0.018
#> Item_2   -0.015   0.025
#> Item_3    0.022   0.022
#> Item_4    0.014  -0.011
#> Item_5    0.017   0.007
#> Item_6   -0.025  -0.013
#> Item_7   -0.030  -0.015
#> Item_8   -0.017  -0.012
#> Item_9   -0.035  -0.013
#> Item_10   0.067   0.091
#> Item_11  -0.017  -0.009
#> Item_12  -0.015  -0.006
#> Item_13  -0.027  -0.048
#> Item_14  -0.032  -0.028
#> Item_15  -0.032   0.014
#> Item_16  -0.024  -0.009
#> Item_17  -0.064  -0.047
#> Item_18  -0.014  -0.029
#> Item_19  -0.047  -0.019
#> Item_20  -0.018  -0.050
#> Item_21  -0.023  -0.019
#> Item_22  -0.030   0.016
#> Item_23  -0.033   0.037
#> Item_24  -0.042  -0.039
#> Item_25  -0.025   0.016
#> Item_26  -0.030   0.023
#> Item_27  -0.048   0.007
#> Item_28  -0.040  -0.035
#> Item_29          -0.029
#> Item_30   2.120        

# EAP estimates
fscores(mltm) |> head()
#>           theta1     theta2
#> [1,] -2.00196078 -0.4814752
#> [2,]  1.13844928  0.3697978
#> [3,]  0.14931661  1.7928628
#> [4,] -1.30231356 -0.2349522
#> [5,] -0.09142943  1.4551918
#> [6,]  1.47204612  1.1222933

# }